107. let p(x,y,z) be a point situated at an equal distance from points a(1,−1,0) and b(−1,2,1). show that point p lies on the plane of equation −2x 3y z

The AA Similarity Postulate only considers angle congruence, the SSS Similarity Theorem compares the ratios of all three pairs of corresponding sides, and the SAS Similarity Theorem considers the ratio of two sides and the congruence of the included angle. These principles provide different criteria for determining similarity between triangles.

The AA Similarity Postulate, the SSS Similarity Theorem, and the SAS Similarity Theorem are all principles used in geometry to determine if two figures are similar. While they serve similar purposes, there are differences in the conditions required for similarity.

AA Similarity Postulate:

The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. In other words, if the corresponding angles of two triangles are equal, the triangles are similar. This postulate does not require any specific information about the side lengths.

SSS Similarity Theorem:

The SSS Similarity Theorem states that if the ratios of the corresponding side lengths of two triangles are equal, then the triangles are similar. This theorem requires that all three pairs of corresponding sides have proportional lengths. In other words, if the lengths of the corresponding sides of two triangles are in proportion, the triangles are similar.

SAS Similarity Theorem:

The SAS Similarity Theorem states that if the ratio of the lengths of two pairs of corresponding sides of two triangles is equal, and the included angles between those sides are congruent, then the triangles are similar. This theorem requires that two pairs of corresponding sides are proportional in length and the included angles are congruent.

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The factors of the expression are (90x - 100)(90x + 100).

The expression 8100x² - 10,000 can be factored completely as the difference of squares. The factored form is (90x - 100)(90x + 100).

To factor the given expression, we can recognize that 8100x² is a perfect square, as it can be expressed as (90x)². Similarly, 10,000 is also a perfect square, as it can be expressed as (100)².

Using the difference of squares formula, which states that a² - b² can be factored as (a + b)(a - b), we can rewrite the expression as (90x)² - (100)².

Applying the difference of squares formula, we have (90x - 100)(90x + 100).

Therefore, the completely factored form of 8100x² - 10,000 is (90x - 100)(90x + 100).

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An ellipse with an eccentricity close to 0 is very close to being a perfect circle.

When the eccentricity of an ellipse is close to 0, it means that the distance between the center and the foci (c) is almost equal to the distance between the center and the vertices (a). In other words, the foci are very close to the center of the ellipse.

In a perfect circle, the foci and the center coincide, and the distance from the center to any point on the boundary (the radius) is always the same. As the eccentricity approaches 0, the ellipse becomes more and more similar to a circle, with the foci getting closer to the center.

Therefore, an ellipse with an eccentricity close to 0 will have a shape that closely resembles a circle.

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When i = 0.13 and N = 7, the evaluated value of the equation [tex]i(1+i)^(N/(1+i)[/tex]) is approximately 0.2517.

To evaluate the equation[tex]i(1+i)^(N/(1+i))[/tex], where i = 0.13 and N = 7. we can substitute these values into the equation and calculate the result.

[tex]i(1+i)^(N/(1+i))[/tex] = 0.13(1 + 0.13)^(7/(1 + 0.13))

Calculating the values inside the parentheses first:

1 + 0.13 = 1.13

Now we can substitute these values into the equation:

[tex]0.13 * (1.13)^(7/1.13)[/tex]

Using a calculator or software to perform the calculations, we find:

0.13 * (1.13)^(7/1.13) ≈ 0.13 * 1.9379 ≈ 0.2517

Therefore, when i = 0.13 and N = 7, the evaluated value of the equation [tex]i(1+i)^(N/(1+i)[/tex]) is approximately 0.2517.

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a.The interest rate earned when borrowing $650 is approximately 12 percent. b. The interest rate earned when borrowing $650 is approximately 12 percent. c.The interest rate earned when borrowing $74,000 is approximately 6 percent,. d. The interest rate earned when borrowing $18,000  is approximately 8 percent. .

a. The interest rate earned when borrowing $650 and repaying $728 after 1 year is approximately 12 percent. b. The interest rate earned when lending $650 and receiving $728 after 1 year is also approximately 12 percent. c. The interest rate earned when borrowing $74,000 and repaying $127,146 after 8 years is approximately 6 percent. d. The interest rate earned when borrowing $18,000 and making payments of $4,390.00 annually for 5 years is approximately 8 percent.

To calculate the interest rate earned in each scenario, we can use the formula for compound interest. The formula is:

Future Value = Present Value × (1 + Interest Rate)^Number of Periods

Rearranging the formula, we can solve for the interest rate:

Interest Rate = ((Future Value / Present Value)^(1 / Number of Periods) - 1) × 100

By plugging in the given values and solving for the interest rate, we can determine the approximate interest rates earned in each case. The interest rates are rounded to the nearest whole number.

For example, in scenario a, the interest rate earned is calculated as ((728 / 650)^(1/1) - 1) × 100, which results in approximately 12 percent. This means that by borrowing $650 and repaying $728 after 1 year, you would be earning an interest rate of around 12 percent. Similarly, the interest rates for scenarios b, c, and d can be calculated using the same formula to obtain the respective answers.

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The probability of getting an odd or a number greater than 2 when tossing a standard number cube is 1.

The probability of getting an odd or a number greater than 2 when tossing a standard number cube can be found as follows:

P(odd or greater than 2) = P(odd) + P(greater than 2) - P(odd and greater than 2)

The probability of getting an odd number is 3 out of 6, since there are three odd numbers (1, 3, 5) on a standard number cube. Therefore, P(odd) = 3/6 = 1/2.

The probability of getting a number greater than 2 is 4 out of 6, as there are four numbers (3, 4, 5, 6) greater than 2 on a standard number cube. Hence, P(greater than 2) = 4/6 = 2/3.

To find the probability of getting both an odd number and a number greater than 2, we need to determine the number of outcomes that satisfy both conditions. There is only one number that satisfies both conditions, which is 3. Therefore, P(odd and greater than 2) = 1/6.

Now, we can substitute the values into the formula:

P(odd or greater than 2) = P(odd) + P(greater than 2) - P(odd and greater than 2)

P(odd or greater than 2) = 1/2 + 2/3 - 1/6

To simplify the expression, we need to find a common denominator for the fractions:

P(odd or greater than 2) = 3/6 + 4/6 - 1/6

P(odd or greater than 2) = 6/6

P(odd or greater than 2) = 1

Therefore, the probability of getting an odd or a number greater than 2 when tossing a standard number cube is 1.

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The values at the 30th and 90th percentiles for the given data set are 5896 and 6283, respectively.

To find the values at the 30th and 90th percentiles for the given data set, we can follow these steps:
1. Sort the data set in ascending order:
 5581  5700  5700  5896  5972  5993  6075  6274  6283  6381

2. Calculate the indices for the 30th and 90th percentiles:
   30th percentile index = (30/100) * (n+1)
   90th percentile index = (90/100) * (n+1)
   where n is the total number of data points.
3. Determine the values at the calculated indices:
   For the 30th percentile, the index is (30/100) * (10+1) = 3.3, which rounds up to 4. Therefore, the value at the 30th          percentile is the 4th value in the sorted data set, which is 5896.
For the 90th percentile, the index is (90/100) * (10+1) = 9.9, which rounds up to 10. Therefore, the value at the 90th percentile is the 10th value in the sorted data set, which is 6283.

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Based on the given information and the Mean Value Theorem, we can determine that the largest possible value of f(5) is 21. The Mean Value Theorem guarantees the existence of a point within the interval (−1, 5)


To find the largest possible value of f(5) using the Mean Value Theorem, we can consider the interval [−1, 5]. Since f(x) is continuous on this interval and differentiable on the open interval (−1, 5), the Mean Value Theorem guarantees the existence of a point c in the interval (−1, 5) such that the derivative of f(x) at that point is equal to the average rate of change of f(x) over the interval [−1, 5].

Since f(−1) = −3 and f(x) is continuous on the interval [−1, 5], by the Mean Value Theorem, there exists a point c in the interval (−1, 5) such that f'(c) is equal to the average rate of change of f(x) over the interval [−1, 5]. The average rate of change of f(x) over this interval is given by (f(5) - f(−1))/(5 - (−1)) = (f(5) + 3)/6.

Now, since we are given that f′(x) ≤ 4 for all values of x, we can conclude that f'(c) ≤ 4. Therefore, we have f'(c) ≤ 4 ≤ (f(5) + 3)/6. By rearranging the inequality, we get 24 ≤ f(5) + 3. Subtracting 3 from both sides gives 21 ≤ f(5), which means the largest possible value of f(5) is 21.

By considering the given conditions, such as f(−1) = −3 and f′(x) ≤ 4, we can derive the inequality 21 ≤ f(5) as the largest possible value.

#Let the function f be continuous and differentiable for all x. Suppose you are given that f(−1)=−3, and that f

′ (x)≤4 for all values of x. Use the Mean Value Theorem to determine the largest possible value of f(5).

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The least value of the numbers is (c) 36

From the question, we have the following parameters that can be used in our computation:

Numbers = 3

Both numbers = 1/3 of the greatest

using the above as a guide, we have the following:

x + x + 3x = 180

So, we have

5x = 180

Divide

x = 36

Hence, the least number is 36

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A. The expression 25c2 evaluates to 300.

B. To evaluate the expression 25c2, we need to calculate the value of 25 multiplied by the binomial coefficient 2.

The binomial coefficient, denoted as "c" or sometimes represented by "C" or "choose," is a mathematical function that calculates the number of ways to choose a certain number of items from a larger set.

The binomial coefficient can be calculated using the formula:

nCk = n! / (k!(n-k)!)

In this case, we have 25C2, which means we need to calculate the number of ways to choose 2 items from a set of 25 items.

Plugging the values into the formula, we have:

25C2 = 25! / (2!(25-2)!)

     = 25! / (2! * 23!)

Calculating the factorials, we have:

25! = 25 * 24 * 23!

2! = 2 * 1

Substituting the values back into the equation, we get:

25C2 = (25 * 24 * 23!) / (2 * 1 * 23!)

Simplifying the expression, we find:

25C2 = 25 * 12

     = 300

Therefore, the expression 25c2 evaluates to 300 when expressed using the usual format for writing numbers.

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The expression (x-2) 180 can be evaluated by substituting the given value of x, which is 8 is 1800. That is, the value of the algebraic expression (x-2) 180 is 1080.

The expression (x-2) 180 can be evaluated by substituting the given value of x, which is 8, and following a step-by-step process.

To evaluate the expression (x-2) 180, we substitute the value of x, which is 8.

By simplifying the expression, we first subtract 2 from 8, resulting in 6. Then, we multiply 6 by 180 to obtain the final answer of 1080. The key steps involved are substitution, simplification, and multiplication.

Step 1: Substitute the value of x in the expression: (8-2) 180.

Step 2: Simplify the expression: (6) 180.

Step 3: Perform the multiplication: 1080.

Therefore, when x is equal to 8, the value of the expression (x-2) 180 is 1080.

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a) The utility depends on both health (H) and consumption of other goods (X).

b) The coefficient is negative, indicating a negative relationship between two variables.

c) Health (H) is greater than zero, suggesting a positive value for health.

d) Health (H) is a function of a variable denoted as 'm'.

e) The variable 'm' is greater than zero and the coefficient is negative.

f) The product of two variables, 'm' and 'm', is negative.

a) The expression in (a) is reasonable as it acknowledges that utility is influenced by both health and consumption of other goods. It recognizes that happiness or satisfaction is derived not only from health but also from other aspects of life.

b) The expression in (b) suggests a negative coefficient and a negative relationship between the variables. This could imply that an increase in one variable leads to a decrease in the other. The reasonableness of this relationship would depend on the specific variables involved and the context of the theoretical framework.

c) The expression in (c) states that health (H) is greater than zero, which is reasonable as health is generally considered a positive attribute that contributes to well-being.

d) The expression in (d) indicates that health (H) is a function of a variable denoted as 'm'. The specific nature of the function or the relationship between 'm' and health is not provided, making it difficult to assess its reasonableness without further information.

e) The expression in (e) states that the variable 'm' is greater than zero and the coefficient is negative. This implies that an increase in 'm' leads to a decrease in some other variable. The reasonableness of this relationship depends on the specific variables involved and the theoretical context.

f) The expression in (f) suggests that the product of two variables, 'm' and 'm', is negative. This implies that either 'm' or 'm' (or both) are negative. The reasonableness of this expression would depend on the meaning and interpretation of the variables involved in the theoretical framework.

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(a) Mathematically, this can be expressed as: MRS = p1/p2, where MRS is the marginal rate of substitution and p1/p2 is the price ratio of the two goods. (b) This condition ensures that the consumer would not be willing to trade more units of the second good for the first good at the given prices, as it would violate the optimality condition for utility maximization.

(a) To show that the indifference curve through an interior solution, denoted as x*, must be tangent to the consumer's budget line, we can use the concept of marginal rate of substitution (MRS) and the slope of the budget line.

The MRS measures the rate at which a consumer is willing to trade one good for another while remaining on the same indifference curve. It represents the slope of the indifference curve.

The budget line represents the combinations of goods that the consumer can afford given their income and prices. Its slope is determined by the price ratio of the two goods.

If x* is an interior solution, it means that the consumer is consuming positive amounts of both goods. At x*, the MRS must be equal to the price ratio for the consumer to be in equilibrium.

Mathematically, this can be expressed as:

MRS = p1/p2

where MRS is the marginal rate of substitution and p1/p2 is the price ratio of the two goods.

(b) If x* ∈ [tex]R+^2[/tex]and x1* = 0, it means that the consumer is consuming only the second good and not consuming any units of the first good.

In this case, the marginal utility of the second good (MU2) divided by the marginal utility of the first good (MU1) should be less than the price ratio of the two goods (p2/p1) for the consumer to be in equilibrium.

Mathematically, this can be expressed as:

MU2/MU1 < p2/p1

This condition ensures that the consumer would not be willing to trade more units of the second good for the first good at the given prices, as it would violate the optimality condition for utility maximization.

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Answer:

[tex]\begin{aligned} \textsf{(a)} \quad f(g(x))&=\boxed{x}\\g(f(x))&=\boxed{x}\end{aligned}\\\\\textsf{\;\;\;\;\;\;\;\;$f$ and $g$ are inverses of each other.}[/tex]

[tex]\begin{aligned} \textsf{(b)} \quad f(g(x))&=\boxed{x}\\g(f(x))&=\boxed{x}\end{aligned}\\\\\textsf{\;\;\;\;\;\;\;\;$f$ and $g$ are inverses of each other.}[/tex]

Step-by-step explanation:

Given functions:

[tex]\begin{cases}f(x)=-\dfrac{x}{2}\\\\g(x)=-2x\end{cases}[/tex]

Evaluate the composite function f(g(x)):

[tex]\begin{aligned}f(g(x))&=f(-2x)\\\\&=-\dfrac{-2x}{2}\\\\&=x\end{aligned}[/tex]

Evaluate the composite function g(f(x)):

[tex]\begin{aligned}g(f(x))&=g\left(-\dfrac{x}{2}\right)\\\\&=-2\left(-\dfrac{x}{2}\right)\\\\&=x\end{aligned}[/tex]

The definition of inverse functions states that two functions, f and g, are inverses of each other if and only if their compositions yield the identity function, i.e. f(g(x)) = g(f(x)) = x.

Therefore, as f(g(x)) = g(f(x)) = x, then f and g are inverses of each other.

[tex]\hrulefill[/tex]

Given functions:

[tex]\begin{cases}f(x)=2x+1\\\\g(x)=\dfrac{x-1}{2}\end{cases}[/tex]

Evaluate the composite function f(g(x)):

[tex]\begin{aligned}f(g(x))&=f\left(\dfrac{x-1}{2}\right)\\\\&=2\left(\dfrac{x-1}{2}\right)+1\\\\&=(x-1)+1\\\\&=x\end{aligned}[/tex]

Evaluate the composite function g(f(x)):

[tex]\begin{aligned}g(f(x))&=g(2x+1)\\\\&=\dfrac{(2x+1)-1}{2}\\\\&=\dfrac{2x}{2}\\\\&=x\end{aligned}[/tex]

The definition of inverse functions states that two functions, f and g, are inverses of each other if and only if their compositions yield the identity function, i.e. f(g(x)) = g(f(x)) = x.

Therefore, as f(g(x)) = g(f(x)) = x, then f and g are inverses of each other.

Answer:

see explanation

Step-by-step explanation:

given f(x) and g(x)

if f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

(a)

f(g(x))

= f(- 2x)

= - [tex]\frac{-2x}{2}[/tex] ( cancel 2 on numerator/ denominator )

= x

g(f(x))

= g(- [tex]\frac{x}{2}[/tex] )

= - 2 × - [tex]\frac{x}{2}[/tex] ( cancel 2 on numerator/ denominator )

= x

since f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

(b)

f(g(x))

= f([tex]\frac{x-1}{2}[/tex] )

= 2([tex]\frac{x-1}{2}[/tex] ) + 1

= x - 1 + 1

= x

g(f(x))

= g(2x + 1)

= [tex]\frac{2x+1-1}{2}[/tex]

= [tex]\frac{2x}{2}[/tex]

= x

since f(g(x)) = g(f(x)) = x

then f(x) and g(x) are inverses of each other

The subtraction of the vectors [2, 2, -1, 6] and [4, -1, 0, 5] results in the vector [-2, 3, -1, 1].

To subtract vectors, we subtract the corresponding components of the vectors.

Given vectors:
A = [2, 2, -1, 6]
B = [4, -1, 0, 5]

Subtracting the corresponding components, we get:
A - B = [2 - 4, 2 - (-1), -1 - 0, 6 - 5]
= [-2, 3, -1, 1]

Therefore, the result of the subtraction is [-2, 3, -1, 1].

The resulting vector [x, y, -1, z] represents the difference between the original vectors in each component.

The specific values of x, y, and z can be obtained by substituting the corresponding components from the subtraction. In this case, x = -2, y = 3, and z = 1.

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Answer:

cost of 1 muffin is £1 , cost of cake is £3

Step-by-step explanation:

setting up the simultaneous equations

2x + y = 5 → (1)

5x + y = 8 → (2)

subtract (1) from (2) term by term to eliminate y

(5x - 2x) + (y - y) = 8 - 5

3x + 0 = 3

3x = 3 ( divide both sides by 3 )

x = 1

substitute x = 1 into either of the 2 equations and solve for y

substituting into (1)

2(1) + y = 5

2 + y = 5 ( subtract 2 from both sides )

y = 3

the cost of a muffin is £1 and the cost of a cake is £3

The values of the six trigonometric functions for the angle determined by the point (-5, 12) are:

sin = 12/13

cos = -5/13

tan = -12/5

csc = 13/12

sec = -13/5

cot = -5/12

To determine the values of the six trigonometric functions for a standard-position angle determined by the point (-5, 12), we can use the coordinates of the point to find the values of the opposite, adjacent, and hypotenuse sides of the right triangle formed by the angle.

The coordinates (-5, 12) correspond to the point in the second quadrant of the Cartesian plane.

Using the Pythagorean theorem, we can find the length of the hypotenuse (r) of the right triangle:

r = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13

Now, we can determine the values of the trigonometric functions:

sine (sin) = opposite/hypotenuse = 12/13

cosine (cos) = adjacent/hypotenuse = -5/13 (since it is in the second quadrant, adjacent side is negative)

tangent (tan) = opposite/adjacent = (12/(-5)) = -12/5

cosecant (csc) = 1/sin = 13/12

secant (sec) = 1/cos = -13/5

cotangent (cot) = 1/tan = (-5/12)

Therefore, the values of the six trigonometric functions for the angle determined by the point (-5, 12) are:

sin = 12/13

cos = -5/13

tan = -12/5

csc = 13/12

sec = -13/5

cot = -5/12

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Answer:

Step-by-step explanation:

We know a complete revolution happens in 360 degrees so let set up a proportion.

1.25/5=  x/360

1/4= x/360

x=90

So this angle was 90 degrees.

Notice that the circumference of a circle is 2pi radians.

So a full revolution takes 2pi radians to occur.

1/4 of that will be pi/2 radians.

So, the radians is pi/2

The measure of the angle A from the sine rule is 52.0 degrees.

The sine rule states that in any triangle:

a / sin(A) = b / sin(B) = c / sin(C)

The ratio of the length of each side of the triangle to the sine of the opposite angle is constant for all three sides. This allows us to solve for unknown side lengths or angles in a triangle when certain information is given.

Thus we have to look at the problem that we have so as to ber able solve it and obtain the angle A.

25/Sin A = 28/Sin 62

A= Sin-1(25Sin62/28)

A = 52.0 degrees

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There are approximately 244.9489406 gigaliters in .0469 cubic miles (mi³).

To convert .0469 cubic miles (mi³) to gigaliters (GL), we need to convert the volume from cubic miles to cubic feet, then to cubic inches, cubic centimeters, milliliters, liters, and finally to gigaliters. Each conversion involves multiplying or dividing by a specific conversion factor.
1 cubic mile (mi³) is equal to 5,280 feet × 5,280 feet × 5,280 feet, which is 147,197,952,000 cubic feet (ft³). Therefore, to convert .0469 mi³ to cubic feet, we multiply it by the conversion factor:
.0469 mi³ × 147,197,952,000 ft³/mi³ = 6,899,617,708.8 ft³
Next, we convert cubic feet to cubic inches. There are 12 inches in a foot, so we multiply the cubic feet value by (12 inches)³:
6,899,617,708.8 ft³ × (12 in)³ = 14,915,215,778,816 cubic inches (in³)
To convert cubic inches to cubic centimeters (cm³), we use the conversion factor of 1 inch = 2.54 centimeters:
14,915,215,778,816 in³ × (2.54 cm/in)³ = 2.449489406 × 10¹⁴ cm³
Next, we convert cubic centimeters to milliliters (mL). Since 1 cm³ is equal to 1 mL, the value remains the same:
2.449489406 × 10¹⁴ cm³ = 2.449489406 × 10¹⁴ mL
To convert milliliters to liters (L), we divide the value by 1,000:
2.449489406 × 10¹⁴ mL ÷ 1,000 = 2.449489406 × 10¹¹ L
Finally, to convert liters to gigaliters (GL), we divide the value by 1 billion:
2.449489406 × 10¹¹ L ÷ 1,000,000,000 = 244.9489406 GL
Therefore, there are approximately 244.9489406 gigaliters in .0469 cubic miles (mi³).

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Approximately 63.8% of 58 is equal to 37.To find the percent of 58 that is represented by 37, we can set up an equation.

Let x represent the unknown percentage we are trying to find.

We can set up the equation:

x% of 58 = 37

To solve for x, we can divide both sides of the equation by 58:

(x/100) * 58 = 37

Dividing both sides by 58:

x/100 = 37/58

To isolate x, we can cross multiply:

58x = 37 * 100

58x = 3700

Dividing both sides by 58:

x = 3700/58

x ≈ 63.8

Therefore, approximately 63.8% of 58 is equal to 37.

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True. The Department of Agriculture defines a food desert as a specific geographic area, known as a census tract, where either 33 percent of the population or 500 people (whichever is less) have limited access to a grocery store.

According to the Department of Agriculture, a food desert is defined as a census tract in which a significant portion of the population, or a minimum of 500 people (whichever is less), live a certain distance away from a grocery store.

The specific distance criterion varies depending on whether it is an urban or rural area. The purpose of this definition is to identify areas where residents have limited access to fresh, healthy, and affordable food options.

Food deserts are considered a significant issue as they can contribute to disparities in nutrition and health outcomes within a community.

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The present value today of four years of college costs starting 18 years from today, assuming a discount rate of 8% compounded annually, is approximately $290,360.

To calculate the present value, we need to discount the future college costs back to the present using the discount rate of 8%. The formula for calculating the present value of a future cash flow is:

Present Value = Future Value / [tex](1 + Discount Rate)^{n}[/tex]

Here, the future value is $100,000 per year for four years, and n is the number of years from today to when the college costs start, which is 18 years. Plugging in these values into the formula, we get:

Present Value = ($100,000 / [tex](1 + 0.08)^{18}[/tex]) + ($100,000 /[tex](1 + 0.08)^{19}[/tex]) + ($100,000 / [tex](1 + 0.08)^{20}[/tex]) + ($100,000 / [tex](1 + 0.08)^{21}[/tex])

Evaluating this expression, we find that the present value today of the four years of college costs is approximately $290,360.

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The sum of the number of faces, vertices, and edges of an octagonal pyramid is 19.

In an octagonal pyramid, the base has 8 faces (sides of the octagon), the apex contributes 1 face, and there are 8 triangular faces connecting the apex to each vertex of the base. So, the total number of faces is 8 + 1 + 8 = 17.

The base of the octagonal pyramid has 8 vertices (each corner of the octagon). Since the apex is a single point, it does not contribute any additional vertices. Therefore, the total number of vertices is 8.

Lastly, the base of the octagonal pyramid has 8 edges (connecting each pair of adjacent vertices of the octagon). Each triangular face connecting the apex to the base contributes 3 edges. So, the total number of edges is 8 + (8 * 3) = 32.

To find the sum, we add the number of faces (17), vertices (8), and edges (32) together: 17 + 8 + 32 = 57. Therefore, the sum of the number of faces, vertices, and edges of an octagonal pyramid is 57.

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The distance between the pair of parallel lines with the given equations is[tex]5.68 units.[/tex]

Let's choose a point on Line 1. For simplicity, let's choose the y-intercept, which is (0, -3).

Now, we can use the formula for the distance from a point (x0, y0) to a line[tex]Ax + By + C = 0:[/tex]

Distance = [tex]\dfrac{(A x_0 + B y_0 + C)}{\sqrt{(A^2 + B^2)}}[/tex]

Substitute  the values, we have:

Distance = [tex]=\dfrac{(\frac{1}{3}) \times 0+ (-1) \times (-3) + 3)} {\sqrt{(\frac{1}{3} )^{2} + (-1)^2}}[/tex]

Simplifying the equation further:

Distance =[tex]\frac{3+3}\sqrt\dfrac{1} {9} +1[/tex]

Rationalizing the denominator:

Distance =[tex]\\dfrac{ 6\times \sqrt[9]{10} }{\sqrt{10} }[/tex]

Finally:

[tex]Distance =\dfrac {6 \times3}{{\sqrt10} }[/tex]

Distance = [tex]\dfrac{18}{\sqrt{10} }[/tex]

So, the distance between the pair of parallel lines is[tex]\dfrac{18}{\sqrt{10} }[/tex], which is approximately [tex]5.68 units.[/tex]

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A. The formula for the arithmetic sequence is (n/2)(2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference.

B. To understand the formula for the arithmetic sequence, let's break it down step by step:

1. The arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

For example, 2, 5, 8, 11 is an arithmetic sequence with a common difference of 3.

2. The sum of an arithmetic sequence can be calculated using the formula Sn = (n/2)(2a + (n-1)d), where Sn represents the sum of the first n terms, a is the first term, and d is a common difference.

3. The formula consists of three parts:

  - (n/2) represents the average number of terms in the sequence. It is multiplied by the sum of the first and last term to account for the sum of the terms in the sequence.

  - 2a represents the sum of the first and last term.

  - (n-1)d represents the sum of the differences between consecutive terms.

4. By multiplying these three parts together, we can find the sum of the arithmetic sequence.

In summary, the formula for the arithmetic sequence, when given the sum of the sequence, is (n/2)(2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference.

This formula allows us to calculate the sum of an arithmetic sequence efficiently.

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The probability that the two individuals will meet is 1/3 or approximately 0.3333.

To determine the probability that the two individuals will meet, we need to consider the time window during which they both remain present.

Let's break down the problem step by step:

Determine the possible arrival times for the first individual:

The first individual arrives randomly between 11:00 am and 11:45 am.

Since they stay for 30 minutes, their departure time will be between (arrival time) and (arrival time + 30 minutes).

Determine the possible arrival times for the second individual:

The second individual arrives randomly between 11:30 am and 12:00 pm.

Since they stay for 15 minutes, their departure time will be between (arrival time) and (arrival time + 15 minutes).

Find the overlapping time range:

To find the window when both individuals are present, we need to identify the overlapping time range between their arrival and departure times.

Calculate the probability of meeting:

The probability of meeting is equal to the length of the overlapping time range divided by the total time available for both individuals.

Given the above information, let's calculate the probability of the two individuals meeting:

The overlapping time range occurs when the first individual arrives before the second individual's departure and the second individual arrives before the first individual's departure. This can be visualized as an intersection of the two time ranges.

The overlapping time range for the two individuals is between 11:30 am and 11:45 am because the first individual arrives at the latest by 11:45 am (allowing for a 30-minute stay) and the second individual leaves at the earliest by 11:45 am (after staying for 15 minutes).

The total time available for both individuals is 45 minutes (from 11:00 am to 11:45 am).

Therefore, the probability of the two individuals actually meeting is:

Probability = (length of overlapping time range) / (total time available)

Probability = 15 minutes / 45 minutes

Probability = 1/3 or approximately 0.3333

Hence, the probability that the two individuals will meet is 1/3 or approximately 0.3333.

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To find the optimal solution, we need to determine the feasible region by graphing the given constraints and identify the corner points where the constraints intersect. However, to provide a numerical solution, we can use a linear programming solver.

Using a linear programming solver, we can input the objective function C = 7x + 9y and the constraints 6x + 8y ≤ 15 and 9x + 8y ≤ 19, along with the non-negativity constraints x ≥ 0 and y ≥ 0. The solver will then calculate the optimal values of x and y that maximize the objective function C.

The optimal values of x and y will depend on the specific values of the constraints, and the resulting values may not be whole numbers. Therefore, rounding the answers to three decimal places will provide the desired level of precision. The linear programming solver will provide the optimal values of x and y that maximize the objective function C.

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The probability that a person who took the test scored between 120 and 160 is approximately 0.6826, which is equivalent to 68%.

To find the probability that a person who took the test scored between 120 and 160, we need to calculate the area under the normal distribution curve between these two scores.

First, let's standardize the scores using the Z-score formula:

Z = (X - μ) / σ

Where:

X = Score

μ = Mean score

σ = Standard deviation

For the lower score of 120:

Z1 = (120 - 140) / 20 = -1

For the upper score of 160:

Z2 = (160 - 140) / 20 = 1

Next, we can use a standard normal distribution table or calculator to find the probability associated with each Z-score.

The probability of a Z-score less than -1 is approximately 0.1587 (from the standard normal distribution table), and the probability of a Z-score less than 1 is approximately 0.8413.

To find the probability between the scores of 120 and 160, we subtract the probability associated with the lower score from the probability associated with the upper score:

P(120 < X < 160) = P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)

= 0.8413 - 0.1587

= 0.6826

Therefore, the probability that a person who took the test scored between 120 and 160 is approximately 0.6826, which is equivalent to 68%.

Therefore, the correct answer is C. 68%.

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The length of the altitude to line segment $\overline{em}$ in triangle $gem$ is 16 units.

Let's denote the length of the altitude to line segment $\overline{em}$ in triangle $gem$ as $h$.

The area of a triangle is given by the formula:

Area = (base * height) / 2

The area common to triangle $gem$ and square $aime$ is 80 square units. Since the base of triangle $gem$ is $\overline{em}$, we have:

80 = (10 * h) / 2

160 = 10h

Solving for $h$, we have:

h = 160 / 10

h = 16

Therefore, the length of the altitude to line segment $\overline{em}$ in triangle $gem$ is 16 units.

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