A right triangle with legs of lengths (x 1) and (2x-2) has an area of 80. what is the length of the shorter leg?

A person can determine if the lines are parallel or skewed by comparing their slopes. Parallel lines would have equal slopes while skewed lines would have unequal slopes.

Parallel lines are those lines that do not meet and they lie on the same plane. Whereas, skewed lines do not lie on the same plane and do not intersect.

So, to determine whether the lines are parallel or skewed one has to look at the slopes, the planes, and whether or not they intersect.

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The next time Ken and Larry get their hair cut on the same day will be on a Tuesday.

To determine the day of the week the next time they get their hair cut on the same day, we need to find the least common multiple (LCM) of 20 and 26. The LCM represents the smallest number that is divisible by both 20 and 26, indicating when the two events will coincide again.

Prime factorizing 20 and 26, we have:

20 = 2^2 * 5

26 = 2 * 13

To find the LCM, we take the highest power of each prime factor that appears in either number:

LCM = 2^2 * 5 * 13 = 260

Since 260 days have passed, we know that Ken and Larry will get their hair cut on the same day again after 260 days.

Now, we need to determine the day of the week after 260 days from the initial Tuesday. We can use the fact that there are 7 days in a week and divide 260 by 7 to find the remainder:

260 ÷ 7 = 37 remainder 1

Since there is a remainder of 1, we need to count one day forward from Tuesday. Therefore, the next time Ken and Larry get their hair cut on the same day will be on a Tuesday again.

Hence, the day of the week the next time they get their hair cut on the same day is Tuesday.

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In the given scenario where line EF is tangent to circle G at point A and the measure of angle EAF is 95°, the measure of angle AFG cannot be determined with the information provided.

More details or measurements are needed to calculate the specific measure of angle AFG.The information given states that line EF is tangent to circle G at point A and the measure of angle EAF is 95°.

However, this information alone is insufficient to determine the measure of angle AFG. The measure of angle AFG depends on the specific measurements or relationships between the angles and segments within the circle.

Without additional information, such as the measurement of another angle or the length of a segment, it is not possible to calculate the measure of angle AFG. Therefore, the specific measure of angle AFG cannot be determined based on the given information.

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i. there are 214 families who watch at least one platform.

ii. there are 71 families who watch exactly one platform.

To find the number of families who watch at least one platform, we need to add up the number of families who watch each platform individually and subtract the number of families who watch all three platforms.

i. The number of families who watch at least one platform:

= (Number of families who watch Netflix) + (Number of families who watch Hulu) + (Number of families who watch Amazon Prime video) - (Number of families who watch all three platforms)

[tex]= 123 + 124 + 110 - 143[/tex]

[tex]= 214[/tex]

Therefore, there are 214 families who watch at least one platform.

ii. The number of families who watch exactly one platform can be found by adding up the number of families who watch each platform individually and subtracting the number of families who watch more than one platform.

ii. The number of families who watch exactly one platform:

= (Number of families who watch Netflix) + (Number of families who watch Hulu) + (Number of families who watch Amazon Prime video) - 2 * (Number of families who watch all three platforms)

[tex]= 123 + 124 + 110 - 2 * 143= 71[/tex]

Therefore, there are 71 families who watch exactly one platform.

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The given equation is in slope-intercept form, on the x-y coordinate plane.

To arrive at a conclusion, we need to understand line equations in Coordinate Geometry.

In 2-D Coordinate Geometry, there are several ways in which the equations of lines can be written. All of these differ according to the parameters used to represent the equation, such as the slope, a single point, two points, intercepts, etc.

But for the same graphical representation, all forms of the line equation are supposed to be the same.

We define all three forms of line equations mentioned in the question:

1. Slope Intercept Form

For a line of slope 'm', which intersects the y-axis at a point (0,b), the line equation is as follows.

y = mx + b

2. Slope Point Form

For a line of slope 'm', which passes through a point (x₁ , y₁), the equation is as follows.

(y - y₁) = m(x - x₁)

3. Standard form:

y = mx + c

where c is a constant.

For the given equation y = (-1/4)x + 9,

If we put in x = 0, we get y = 9, which is the y-intercept of the line.

So, we can conclude that the given equation is in slope-intercept form.

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John's preferences are monotone but not strictly monotone. John's preferences are convex but not strictly convex. John's preferences satisfy the diminishing marginal rate of substitution property.

John's preferences are monotone because the utility function V(B,W) is increasing in both B (the quantity of good B) and W (the quantity of good W). However, they are not strictly monotone since the utility function does not strictly increase with each increment of B or W.

John's preferences are convex because the utility function V(B,W) is a strictly convex function. This can be observed from the positive second derivatives of both B and W in the utility function. However, they are not strictly convex since the utility function is not strictly increasing at an increasing rate.

John's preferences satisfy the diminishing marginal rate of substitution (MRS) property. This can be shown by calculating the MRS, which is given by the ratio of the marginal utility of B to the marginal utility of W (∂V/∂B / ∂V/∂W). In this case, the MRS is 6B/W. As the quantities of B and W increase, the MRS decreases, indicating diminishing marginal utility of B relative to W.

To determine John's optimal bundle, we need information about his income (I). With the given prices (P_B = 5 and P_W = 40), we can set up the consumer's optimization problem by maximizing utility subject to the budget constraint (P_B × B + P_W × W = I). By solving this constrained optimization problem, we can find the specific quantities of B and W that maximize John's utility given his income and prices. However, since information about John's income is not provided, we cannot obtain the exact optimal bundle without this information.

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The Reflexive Property of Angle Congruence states that every angle is congruent to itself.

Evidence:

We should consider a point meant as ∠ABC.

By definition, point coinciding implies that two points have a similar measure. To demonstrate the Reflexive Property of Point Compatibility, we want to show that ∠ABC is consistent with itself.

Since ∠ABC is a similar point, clearly the two sides of the point are indistinguishable. The vertex and the two beams that structure the point are exactly similar in the two cases.

Accordingly, by definition, ∠ABC is consistent with itself.

Consequently, the Reflexive Property of Point Consistency holds, as each point is consistent with itself.

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The missing values in the table are 60, 48, 36, 24, 12, and 0.

A graph representing the relationship is shown below.

In order to use this linear function f(x) = 72 - 12x to determine the missing values in the table, we would have to substitute each of the values of x (x-values) into the linear function and then evaluate as follows;

When the value of x = 1, the linear function is given by;

f(x) = 72 - 12(1)

f(x) = 60.

When the value of x = 2, the linear function is given by;

f(x) = 72 - 12(2)

f(x) = 48.

When the value of x = 3, the linear function is given by;

f(x) = 72 - 12(3)

f(x) = 36.

When the value of x = 4, the linear function is given by;

f(x) = 72 - 12(4)

f(x) = 24.

When the value of x = 5, the linear function is given by;

f(x) = 72 - 12(5)

f(x) = 12.

When the value of x = 6, the linear function is given by;

f(x) = 72 - 12(1)

f(x) = 0.

In conclusion we would use an online graphing tool to plot the relationship between the amount of plant food remaining, f(x), and the number of days that have passed (x) as shown in the image below.

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The key attributes of linear functions are that they have a constant slope and a constant y-intercept. The domain and range of linear functions are all real numbers.

The key features of quadratic functions are that they have a parabolic shape and they have two roots. The domain and range of quadratic functions are all real numbers.

The key attributes of linear functions can be seen in their graph. A linear function graph is a straight line. The slope of the line tells us how much the y-value changes for every change in the x-value. The y-intercept tells us the value of y when x is 0.

The domain and range of linear functions are all real numbers. This means that the x-value and the y-value can be any real number.

The key features of quadratic functions can be seen in their graph. A quadratic function graph is a parabola. The parabola opens up or down depending on the coefficient of the x^2 term. The roots of the quadratic function are the points where the graph crosses the x-axis.

The domain and range of quadratic functions are all real numbers. This means that the x-value can be any real number, but the y-value cannot be less than or equal to 0.

The key attributes of exponential functions are that they have an exponential growth or decay rate and they have an initial value. The domain and range of exponential functions depend on the base of the exponent.

If the base of the exponent is greater than 1, then the function has an exponential growth rate. This means that the y-value increases rapidly as the x-value increases. If the base of the exponent is less than 1, then the function has an exponential decay rate. This means that the y-value decreases rapidly as the x-value increases.

The domain and range of exponential functions depend on the base of the exponent. If the base of the exponent is greater than 1, then the domain is all real numbers and the range is all positive real numbers. If the base of the exponent is less than 1, then the domain is all real numbers and the range is all real numbers less than or equal to 1.

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To write three radical expressions that simplify to -2x², we can use the concept of raising a number to a fractional exponent.

By using fractional exponents, we can express the square root of a number as a power. Here are three possible radical expressions:
1. (-2x²)^(1/2): This expression represents the square root of -2x². When we raise -2x² to the power of 1/2, it simplifies to √(-2x²), which is equal to ±i√(2x²). Here, i represents the imaginary unit.

2. (-2x²)^(2/4): This expression represents the fourth root of -2x². By raising -2x² to the power of 2/4, we can simplify it as (√(-2x²))^2. This further simplifies to (±i√(2x²))^2, which is equal to -2x².

3. (-2x²)^(3/6): This expression represents the sixth root of -2x². Raising -2x² to the power of 3/6 simplifies it as (∛(-2x²))^3. This can be further simplified to (±∛(2x²))^3, which is again equal to -2x². Three radical expressions that simplify to -2x² are (√(-2x²)), (√(-2x²))^2, and (∛(-2x²))^3. These expressions represent different roots (square root, fourth root, and sixth root) of -2x² and all simplify to -2x².

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To estimate the mean IQ of all students with 95% confidence, we should use a confidence interval. Specifically, we can use the formula for a confidence interval for the population mean when the sample standard deviation is known.

The formula for the confidence interval is:

CI = X ± Z * (σ / √n)

Where:

- X is the sample mean (101.8 in this case)

- Z is the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of approximately 1.96)

- σ is the population standard deviation (5.65 in this case)

- n is the sample size (80 in this case)

By plugging in the values, we can calculate the confidence interval.

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The decimal value of the trigonometric function sec(3π/2) is undefined. To find this value, we use the fact that sec(x) = 1/cos(x) and find the value of cos(3π/2), which is 0. Taking the reciprocal, we get 1/0, which is undefined.

To find the decimal value of sec(3π/2), we can use the fact that:

sec(x) = 1 / cos(x)

So we need to find the value of cosine function cos(3π/2) and take the reciprocal.

Recall that the cosine function has a period of 2π and is symmetric about the vertical line x = π/2. Therefore, cos(3π/2) = cos(π/2) = 0.

Taking the reciprocal, we get:

sec(3π/2) = 1 / cos(3π/2) = 1 / 0

Since division by zero is undefined, sec(3π/2) is undefined.

Therefore, the decimal value of sec(3π/2) is undefined.

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Answer:

The quotient is x^4 - 2x^3 - 9x^2 - 3x with a remainder of -34.

Step-by-step explanation:

To divide the polynomial x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34 by x + 8 using long division, we can follow these steps:

         _______________________

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

Step 1: Divide the first term of the dividend (x^5) by the first term of the divisor (x), which gives x^4. Write this as the first term of the quotient above the line.

               x^4

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

Step 2: Multiply the divisor (x + 8) by the quotient term (x^4). Write the result below the dividend, and subtract it from the dividend.

               x^4

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

- (x^5 + 8x^4)

_______________________

7x^4 + 54x^3 - 25x^2 - 75x

Step 3: Bring down the next term of the dividend, which is 54x^3. Now we have a new dividend.

               x^4

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

- (x^5 + 8x^4)

_______________________

7x^4 + 54x^3 - 25x^2 - 75x

- (7x^4 + 56x^3)

_______________________

-2x^3 - 25x^2 - 75x

Step 4: Divide the new first term of the dividend (-2x^3) by the first term of the divisor (x), which gives -2x^2. Write this as the next term of the quotient above the line.

x^4 - 2x^3

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

- (x^5 + 8x^4)

_______________________

7x^4 + 54x^3 - 25x^2 - 75x

- (7x^4 + 56x^3)

_______________________

-2x^3 - 25x^2 - 75x

Step 5: Multiply the divisor (x + 8) by the new quotient term (-2x^3). Write the result below the previous difference, and subtract it from the previous difference.

               x^4 - 2x^3

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

- (x^5 + 8x^4)

_______________________

7x^4 + 54x^3 - 25x^2 - 75x

- (7x^4 + 56x^3)

_______________________

-2x^3 - 25x^2 - 75x

- (-2x^3 - 16x^2)

_______________________

-9x^2 - 75x

Step 6: Bring down the next term of the dividend, which is -34. Now we have a new dividend.

  x^4 - 2x^3

x + 8 | x^5 + 15x^4 + 54x^3 - 25x^2 - 75x - 34

- (x^5 + 8x^4)

_______________________

7x^4 + 54x^3 - 25x^2 - 75x

- (7x^4 + 56x^3)

_______________________

-2x^3 - 25x^2 - 75x

- (-2x^3 - 16x^2)

_______________________

-9x^2 - 75x

- (-9x^2 - 72x)

_______________________

-3x - 34

Step 7: The division is complete. The final result is -3x - 34.

Therefore, the quotient is x^4 - 2x^3 - 9x^2 - 3x with a remainder of -34.

The volume of the secret room is 504 cubic yards.

To calculate the volume of the secret room shaped like a rectangular prism, we multiply its length, width, and height together.

The length of the room is given as 8 yards, the width as 7 yards, and the height as 9 yards.

Volume = Length × Width × Height

Volume = 8 yards × 7 yards × 9 yards

Calculating the multiplication:

Volume = 504 cubic yards

Therefore, the volume of the secret room is 504 cubic yards.

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The result of the division operation is (3/4). The reciprocal of a fraction is obtained by flipping the numerator and denominator.

To perform the division of (4x/5) ÷ (16/15x), we can simplify the expression by multiplying the numerator of the first fraction by the reciprocal of the second fraction.

The given expression can be rewritten as (4x/5) * (15x/16).

To simplify the expression further, we can cancel out common factors between the numerator of the first fraction and the denominator of the second fraction. In this case, we can cancel out a factor of 4 and a factor of 5, which leaves us with (x/1) * (3x/4).

Now we can multiply the numerators together and the denominators together, resulting in (3x^2/4).

Therefore, the final answer is (3x^2/4), or in fractional form, (3/4)x^2.

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The exact value of cos 180° is -1To find the exact value of cos 180° using a half-angle identity, we can use the half-angle formula for cosine .

cos^2(x/2) = (1 + cos(x))/2

Let's substitute x = 180° into the formula:

cos^2(180°/2) = (1 + cos(180°))/2

Simplifying the expression:

cos^2(90°) = (1 + cos(180°))/2

Now, we know that cos(90°) = 0, so we can substitute that value in:

0 = (1 + cos(180°))/2

Multiplying both sides by 2:

0 = 1 + cos(180°)

Rearranging the equation:

cos(180°) = -1

Therefore, the exact value of cos 180° is -1.

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(a) The point on the graph of y = g(x+4)−5 is (10, g(10)-5).

(b) The point on the graph of y = −3g(x−7)+5 is (-1, -3g(-1)+5).

(c) The point on the graph of y = g(3x+15) is (33, g(33)).

(a) To find the point on the graph of y = g(x+4)−5, we substitute x = 6 into the equation and evaluate y:

y = g(6+4) - 5

y = g(10) - 5

Therefore, the point on the graph of y = g(x+4)−5 is (10, g(10)-5).

(b) To find the point on the graph of y = −3g(x−7)+5, we substitute x = 6 into the equation and evaluate y:

y = -3g(6-7) + 5

y = -3g(-1) + 5

Therefore, the point on the graph of y = −3g(x−7)+5 is (-1, -3g(-1)+5).

(c) To find the point on the graph of y = g(3x+15), we substitute x = 6 into the equation and evaluate y:

y = g(3(6)+15)

y = g(18+15)

y = g(33)

Therefore, the point on the graph of y = g(3x+15) is (33, g(33)).

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The x-component of the total electric field at point P is -3.50 x 10^4 N/C and the y-component is 0 N/C.

To find the x- and y-components of the total electric field caused by q1 and q2 at a point (0.200 m, 0), we need to use the equations for the electric field due to a point charge:

E = k*q/r^2

where E is the electric field in N/C, k is Coulomb's constant (9.0 x 10^9 N*m^2/C^2), q is the charge in C, and r is the distance from the point charge to the point where we want to find the electric field.

Let q1 = +5.00 nC and q2 = -3.00 nC be the charges located at (0.100 m, 0) and (-0.100 m, 0), respectively.

The x-component of the electric field at point P due to q1 is given by:

E1x = kq1(x1-xp)/r1^3

where x1 = 0.100 m is the x-coordinate of q1, xp = 0.200 m is the x-coordinate of point P, and r1 is the distance between q1 and P.

r1 = [(xp-x1)^2 + y^2]^0.5 = [(0.200-0.100)^2 + (0)^2]^0.5 = 0.1 m

E1x = (9.0 x 10^9)(5.00 x 10^-9)(0.100-0.200)/(0.1^3) = -4.50 x 10^4 N/C

Similarly, the x-component of the electric field at point P due to q2 is given by:

E2x = kq2(x2-xp)/r2^3

where x2 = -0.100 m is the x-coordinate of q2, and r2 is the distance between q2 and P.

r2 = [(xp-x2)^2 + y^2]^0.5 = [(0.200+0.100)^2 + (0)^2]^0.5 = 0.3 m

E2x = (9.0 x 10^9)(-3.00 x 10^-9)(0.200+0.100)/(0.3^3) = 1.00 x 10^4 N/C

The total x-component of the electric field at point P is:

Etotal,x = E1x + E2x = -3.50 x 10^4 N/C

To find the y-component of the total electric field, we use the same equations but with y-coordinates instead of x-coordinates.

The y-component of the electric field at point P due to q1 is given by:

E1y = kq1y/r1^3

where y = 0 is the y-coordinate of both q1 and point P.

E1y = (9.0 x 10^9)*(5.00 x 10^-9)*0/(0.1^3) = 0 N/C

Similarly, the y-component of the electric field at point P due to q2 is given by:

E2y = kq2y/r2^3

E2y = (9.0 x 10^9)*(-3.00 x 10^-9)*0/(0.3^3) = 0 N/C

The total y-component of the electric field at point P is:

Etotal,y = E1y + E2y = 0 N/C

Therefore, the x-component of the total electric field at point P is -3.50 x 10^4 N/C and the y-component is 0 N/C.

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Answer:

[tex] - 5 \frac{1}{8} [/tex]

Step-by-step explanation:

[tex]1. \: \frac{4 \times 10 + 1}{10} \times - 3 \times \frac{5}{12} \\ 2. \: \frac{40 + 1}{10} \times - 3 \times \frac{5}{12} \\ 3. \: \frac{41}{10} \times - 3 \times \frac{5}{12} \\ 4. \: \frac{41 \times - 3 \times 5}{10 \times 12} \\ 5. \: \frac{ - 123 \times 5}{10 \times 12} \\ 6. \: \frac{ - 615}{10 \times 12} \\ 7. \: \frac{ - 615}{120} \\ 8. \: - \frac{615}{120} \\ 9. \: - \frac{41}{8} \\ 10. \: - 5 \frac{1}{8} [/tex]

0.75 kg is equal to 750,000 mg. To convert kilograms (kg) to milligrams (mg), we need to multiply the given value in kilograms by a conversion factor. In this case, the conversion factor is 1 kg = 1,000,000 mg.

By multiplying 0.75 kg by 1,000,000 mg/kg, we find that 0.75 kg is equal to 750,000 mg. The conversion factor of 1,000,000 mg/kg is derived from the fact that there are 1,000 grams (g) in a kilogram and 1,000 milligrams (mg) in a gram. Therefore, multiplying these conversion factors together gives us 1,000,000 mg/kg. In this context, the conversion allows us to express a given mass of 0.75 kg in a smaller unit of measurement, which is milligrams. Milligrams are commonly used for precise measurements or when dealing with very small quantities. So, by converting 0.75 kg to 750,000 mg, we have a representation of the same mass in a more granular unit, which can be useful in certain scientific or technical calculations.

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The solution of number is,

⇒ √-15 = √15i

We have to give that,

Simplify the number by using the imaginary number i.

⇒ √-15

Since We know that,

⇒ i² = - 1

⇒ i = √- 1

Hence, We can simplify it as,

⇒ √-15

⇒ √15 × √- 1

⇒ √15 × i

⇒ √(15)i

Therefore, The solution is,

⇒ √-15 = √15i

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There are 720 possible permutations of three digits randomly selected from 0 to 9 without repetition.

To find the possible permutations of three digits randomly selected from 0 to 9, we can use the concept of permutations. In this case, we have 10 digits to choose from (0 to 9), and we want to select three digits without repetition.

The following is the formula to calculate permutations:

P(n, r) = n! / (n - r)!

Where r is the number of items to select, and n is the total number of items from which to choose.

Using this formula, let's calculate the number of permutations for this scenario:

P(10, 3) = 10! / (10 - 3)!

P(10, 3) = 10! / 7!

P(10, 3) = (10 * 9 * 8 * 7* 6* 5* 4* 3* 2* 1) / 7!

The 7! terms cancel out, leaving us with:

P(10, 3) = 10 * 9 * 8

P(10, 3) = 720

Therefore, there are 720 possible permutations of three digits randomly selected from 0 to 9 without repetition.

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From the given data, using the relative frequencies and time range, we can say out of the 100 sampled calls, 10 calls would have lasted at least 15 minutes but less than 20 minutes.

In order to calculate the number calls that would have lasted within the given range, we can add the relative frequencies of the time range.

Given the following information:

Time Range      Relative Frequency

0 < t < 5                      0.37

5 < t < 10                    0.22

10 < t < 15                   0.15

15 < t < 20                  0.10

20 < t < 25                 0.07

25 < t < 30                 0.07

t ≥ 30                         0.02

From the time range of the relative frequencies, this shows that 10% of the 100 sampled calls would last at least 15 minutes but less than 20 minutes.

To calculate the actual number of calls, we multiply the relative frequency by the total number of calls:

Number of calls = Relative frequency * Total number of calls

Number of calls = 0.10 * 100

Number of calls = 10

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The factored form of the expression 2x² + 13x - 7 is (2x + 7)(x - 1).

To factor the expression 2x² + 13x - 7, we need to find two binomials that, when multiplied, result in the given expression.

The first term of each binomial will have the factors of 2x², which can be written as (2x)(x) or (x)(2x).

The last term of each binomial will have the factors of -7, which can be written as (-7)(1) or (1)(-7).

Now, we need to find the factors of -7 that add up to the coefficient of the middle term, which is 13x. The factors of -7 are -7 and 1, and their sum is 13. So, we can rewrite the middle term as 13x as (7x + 1x).

Putting it all together, we have:

2x² + 13x - 7 = (2x + 7)(x - 1)

Therefore, the factored form of the expression 2x² + 13x - 7 is (2x + 7)(x - 1).

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The circled section represents the range of solving times for children and adults at a Rubik's cube competition, indicating variability in performance.

The circled section represents the range of times it took both children and adults to solve a Rubik's cube at a competition. The range is the difference between the highest and lowest times recorded for each group.

It provides an overview of the variability in solving times within each age group. In competitions, participants are timed while solving the cube, and the circled section helps visualize the spread of solving times.

A larger circled section indicates a wider range of solving abilities within the group, while a smaller circled section suggests a more consistent performance.

By examining the circled section, one can gain insights into the skill levels and proficiency of both children and adults in solving Rubik's cubes at the competition.

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To calculate the mass of the wire c with density (x, y) = xy, where c: r(t) = 4t i + 3t j with 0 ≤ t ≤ 2, we need to integrate the product of the density function and the magnitude of the velocity vector along the curve.

The wire c is represented by the parametric equation r(t) = 4t i + 3t j, where i and j are the unit vectors in the x and y directions, respectively, and t is the parameter.

To calculate the mass of the wire, we need to integrate the product of the density function (x, y) = xy and the magnitude of the velocity vector |r'(t)| along the curve.

The velocity vector r'(t) is obtained by differentiating r(t) with respect to t. In this case, r'(t) = 4i + 3j.

The magnitude of the velocity vector |r'(t)| is given by √(4^2 + 3^2) = √25 = 5.

The integral for calculating the mass is given by:

M = ∫[0,2] (density function) * |r'(t)| dt

  = ∫[0,2] xy * 5 dt

  = 5 ∫[0,2] xy dt.

To evaluate this integral, we need additional information about the density function, such as its relationship to the position vector r(t). Without knowing the specific form of the density function, we cannot determine the exact value of the mass of the wire.

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The face value of the non-interest-bearing note for the woman's loan should be approximately $500.26. The face value of the non-interest-bearing note for the man's loan should be approximately $2417.91.

To find the face value of a non-interest-bearing note, we can use the formula:

Face Value = Proceeds / (1 - Discount Rate * (Days to Maturity / 360))

1. Calculation for the woman's loan:

Proceeds = $485

Discount Rate = 13.22%

Days to Maturity = 80 (from April 10 to June 29)

Face Value = $485 / (1 - 0.1322 * (80 / 360))

Face Value = $485 / (1 - 0.1322 * 0.2222)

Face Value = $485 / (1 - 0.0294)

Face Value = $485 / 0.9706

Face Value = $500.26 (approximately)

Therefore, the face value of the non-interest-bearing note for the woman's loan should be approximately $500.26.

2. Calculation for the man's loan:

Amount needed = $2350

Discount Rate = 14.4%

Days to Maturity = 70 (from July 10 to September 18)

Face Value = $2350 / (1 - 0.144 * (70 / 360))

Face Value = $2350 / (1 - 0.144 * 0.1944)

Face Value = $2350 / (1 - 0.0279)

Face Value = $2350 / 0.9721

Face Value = $2417.91 (approximately)

Therefore, the face value of the non-interest-bearing note for the man's loan should be approximately $2417.91.

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(sin θ)(cot θ) is equal to 9/20, which is approximately 0.45 when rounded to the nearest tenth.

To find (sin θ)(cot θ), we need to express cot θ in terms of sin θ.

Recall that cot θ is the reciprocal of tan θ. Since tan θ = 4/3, we can find cot θ by taking the reciprocal:

cot θ = 1/(tan θ) = 1/(4/3) = 3/4

Now, we can substitute sin θ and cot θ into the expression (sin θ)(cot θ):

(sin θ)(cot θ) = (sin θ)(3/4)

To find sin θ, we can use the Pythagorean identity:

sin θ = √(1 - cos² θ)

Given that -π/2 ≤ θ < π/2, we know that cos θ is positive.

Using the identity sin θ = √(1 - cos² θ), we have:

sin θ = √(1 - (cos θ)²)

= √(1 - (4/5)²) [Since cos θ = 4/5, based on the given value of tan θ]

= √(1 - 16/25)

= √(9/25)

= 3/5

Substituting sin θ = 3/5 and cot θ = 3/4 into (sin θ)(cot θ):

(sin θ)(cot θ) = (3/5)(3/4)

= 9/20

Therefore, (sin θ)(cot θ) is equal to 9/20, which is approximately 0.45 when rounded to the nearest tenth.

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The given sequence 16, 7, -2 is arithmetic progression with a common difference of -9. The next term in the sequence is found by adding the common difference. Therefore, the complete sequence is 16, 7, -2, -11, ....

To determine whether the sequence 16, 7, -2, ... is arithmetic, we need to check if there is a common difference between consecutive terms.

The common difference (d) between consecutive terms of an arithmetic sequence is given by:

d = a(n) - a(n-1)

where a(n) is the nth term of the sequence.

From the given terms, we can see that:

a(1) = 16

a(2) = 7

a(3) = -2

Using the formula for the common difference, we have:

d = a(2) - a(1) = 7 - 16 = -9

d = a(3) - a(2) = -2 - 7 = -9

Since the common difference is the same for both pairs of consecutive terms, we can conclude that the sequence is arithmetic with a common difference of -9.

To find the next term in the sequence, we can add the common difference to the previous term:

a(4) = a(3) + d = -2 - 9 = -11

Therefore, the complete sequence is:

16, 7, -2, -11, ..

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Answer:

length = 13 in, width = 7 in , area = 91 in²

Step-by-step explanation:

let width be w then length = 2w - 1

perimeter(P) of rectangle is calculated as

P = 2 × length + 2× width

  = 2(2w - 1) + 2w

  = 4w - 2 + 2w

  = 6w - 2

given P = 40 , then

6w - 2 = 40 ( add 2 to both sides )

6w = 42 ( divide both sides by 6 )

w = 7

then

width = 7 in and length = 2w - 1 = 2(7) - 1 = 14 - 1 = 13 in

the area (A) of a rectangle is calculated as

A = length × width = 13 × 7 = 91 in²