. What is the nature of the image formed by a concave mirror if the magnification produced
by the mirror is:
(a) +3 (b) -1
Predict the size of image in both the cases

An equation for the transformed logarithm shown below, that passes through points (-1, 0) and (4, -2) is f(x) = -2log₆(x + 2).

By critically observing the logarithmic graph that models the logarithm function, we can logically deduce that it was reflected across the y-axis.

In Mathematics, the general form of a logarithm function is represented by the following mathematical equation:

f(x) = -alog(x) + k

Where:

Note: The vertical asymptote is at x = -2. Therefore, the logarithm function is given by:

f(x) = -alog(x + 2) + k

Next, we would determine the values of k and a by using the points (-1, 0) and and (4, -2):

0 = -alog(-1 + 2) + k

0 = -a(0) + k

k = 0.

For the value of a, we have:

-2 = -alog(4 + 2) + 0

-2 = -alog(6)

a = 2/log(6)

In conclusion, the required logarithm function in compact form is given by:

f(x) = -alog(x + 2) + 0

f(x) = -2/log(6) × log(x + 2)

[tex]f(x)=\frac{-2log(x+2)}{log6}[/tex]

f(x) = -2log₆(x + 2)

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The value of 7.682/0.072 using long division is 106 25/36

Division is a mathematical operation which involves the sharing of an amount into equal-sized groups.

By solving this, we need to make both the numerator and denominator a whole number by multiplying all by 1000

= (7.682×1000)/0.072 × 1000

= 7682/72

divide through by 2

= 3841/36

Now there is no common factor between the numerator and denominator,

We therefore find the number of 36 in 3841

= 106 remainder 25

therefore 7.682/0.072

= 106 25/36

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Answer:

Step-by-step explanation:

The absolute value function for this problem is given as follows:

y = |x - 4| - 6.

An absolute value function of vertex (h,k) is defined as follows:

y = a|x - h| + k.

The coordinates of the vertex of the function are given as follows:

(4, -6).

The slope of 1 determines the leading coefficient a as follows:

a = 1.

Hence the function is given as follows:

y = |x - 4| - 6.

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The equation has one real solution when n = 43/9.

To find the values of n for which the equation has one real solution, we need to analyze the discriminant of the quadratic equation. The discriminant is the expression inside the square root of the quadratic formula and determines the nature of the solutions.

In this case, the equation is:

8s = -(n-3)s^2 - 9

To analyze the discriminant, we consider the quadratic equation in the form of as^2 + bs + c = 0. Comparing it with our equation, we have a = -(n-3), b = 8, and c = -9.

The discriminant (D) can be calculated using the formula D = b^2 - 4ac. Substituting the values, we get:

D = 8^2 - 4*(-(n-3))*(-9)

D = 64 - 36(n-3)

D = 64 + 36(3 - n)

D = 64 + 108 - 36n

D = 172 - 36n

For the equation to have one real solution, the discriminant must be equal to zero, D = 0. So, we have:

172 - 36n = 0

36n = 172

n = 172/36

n = 43/9

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The given equation are linear equations.

To determine whether the given system of equations is linear or nonlinear, we need to examine the power of the variables in each equation.

98317y=1010+2965x

-71389y=5692x-1001

The variable y appears to the power of 1 (no exponent).

The variable x appears to the power of 1 (no exponent).

Since both variables have a power of 1 in both equations, the system is a linear system of equations.

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Answer:

5x+7 > 17 matches with 3.

1+3x > -5 matches with 2.

2x+1 < 3 matches with 1.

5-3x > 11 matches with 4.

Step-by-step explanation:

The leg Adjacent the angle denoted as A using the principle of trigonometry is b.

The longest leg of a right angle triangle is the hypotenus. The leg which is directly opposite any given angle is the opposite leg while the leg parallel to any given angle is the Adjacent leg.

From the triangle given, the leg parallel to AC is the adjacent leg. Which is the leg b. So, leg b is the Adjacent.

Therefore, the adjacent leg is the one denoted as b.

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The equations are written as follows

For Bert's Cab Company:: y = 1 + 3a

For Madeline's Cab Company: y = 3 + 2a

i. To represent the cost in dollars (y) for a cab ride of a miles for each cab driver, we can write the following system of linear equations:

For Bert's Cab Company:

y = 1 + 3a

For Madeline's Cab Company:

y = 3 + 2a

Here, 'a' represents the distance in miles.

ii. To find the distance at which the cost will be the same for both companies, we can set the two equations equal to each other and solve for 'a':

1 + 3a = 3 + 2a

a = 2

Therefore, the cost will be the same for both companies at a distance of 2 miles.

iii. To determine which cab driver's charge will be less for a ride that is 10 miles in distance, we can substitute 'a = 10' into the equations and compare the resulting costs:

For Bert's Cab Company:

y = 1 + 3(10) = 1 + 30 = 31 dollars

For Madeline's Cab Company:

y = 3 + 2(10) = 3 + 20 = 23 dollars

From the calculations, we can see that Madeline's cab company will charge less for a 10-mile ride, with a cost of $23 compared to Bert's cab company, which charges $31.

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Answer:

about 0.52 of the field

30=30 -75=80 = 50 near 52

Step-by-step explanation:

about 0.52 of the field

about 0.64 of the field

about 0.71 of the field

about 0.87 of the field

120.61 ft³ is the volume of the given cylinder.

Given that the diameter of the construction pipe is 3 ft, the radius (r) would be half of the diameter, which is 1.5 ft.

The length or height (h) of the pipe is given as 17 ft.

Plugging these values into the volume formula:

Volume = 3.14159 * (1.5 )² * 17

Calculating the expression:

Volume = 3.14159 * 2.25 * 17

Volume = 120.61 ft³

Therefore, the volume of the cylindrical construction pipe is approximately 120.645 cubic feet.

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Among the given options, the equation that matches this relationship is B. y = x + 5.

The table shows that as the value of x increases from 1 to 5, the corresponding values of y are 6, 7, 8, 9, and 10.

To determine the equation that represents the relationship between x and y, we can examine the patterns and relationships in the table.

Looking at the values, we can see that the y-values are always 5 more than the x-values.

Therefore, the equation that represents the relationship between x and y is:

y = x + 5

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The statement means that the set X/== is a partition of X, where X is a set and == is an equivalence relation on X.

Intuitively, this means that every element of X belongs to exactly one equivalence class, and the union of all equivalence classes is X.

More formally, a partition of a set X is a set of subsets of X such that:

The union of all subsets is X.

No two subsets have any elements in common.

In this case, the set X/== is a partition of X because it satisfies both of these conditions. First, the union of all equivalence classes is X because every element of X belongs to exactly one equivalence class.

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There is approximately an 89.44% probability that a box of cereal is underfilled by 5 g or more.

To find the probability that a box of cereal is underfilled by 5 g or more, we need to calculate the probability of obtaining a weight measurement below 345 g.

First, we can standardize the problem by using the z-score formula:

z = (x - μ) / σ

Where:

x = the weight value we want to find the probability for (345 g in this case)

μ = the mean weight (350 g)

σ = the standard deviation (4 g)

Substituting the values into the formula:

z = (345 - 350) / 4 = -1.25

Next, we can find the probability associated with this z-score using a standard normal distribution table or a statistical calculator.

The probability of obtaining a z-score less than -1.25 is approximately 0.1056.

However, we are interested in the probability of underfilling by 5 g or more, which means we need to find the complement of this probability.

The probability of underfilling by 5 g or more is 1 - 0.1056 = 0.8944, or approximately 89.44%.

Therefore, there is approximately an 89.44% probability that a box of cereal is underfilled by 5 g or more.

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The quotient equation from the synthetic division table is (3x⁴ + 19x³ + 22x² - 6x + 4)/(x + 2) = 3x³ + 13x² - 4x + 2

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

The synthetic division table

The synthetic division table has already been completed,

However, we have the following labels

Divisor  |   Dividend

             |  Some numbers

              |  Quotient  Remainder

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

Divisor = x + 2

Quotient = 3x³ + 13x² - 4x + 2

DIvidend = 3x⁴ + 19x³ + 22x² - 6x + 4

Remainder = 0

So, the quotient equation is

(3x⁴ + 19x³ + 22x² - 6x + 4)/(x + 2) = 3x³ + 13x² - 4x + 2

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Answer:

The triangles are similar.

Explanation:

Similar triangles have 3 pairs of congruent angles. In this diagram, we can see that the triangles WXP and WYZ share angle W, which is congruent to itself; therefore, they have at least 1 pair of congruent angles. We are given that angle X is congruent to angle Y, so that is a second pair of congruent angles. Finally, we know that angle P is congruent to angle Z because if two pairs of corresponding angles are congruent, then the third pair must also be congruent because the measures of the interior angles of both triangles have to add to 180°.

Answer:

45 tiles

Step-by-step explanation:

Nan has 30 tiles.

Sato has half the amount Nan does, so you would divide the amount Nan has by two.

30/2=15

Then, you add the two amounts up to find the total amount.

30+15=45 tiles

The expression for the area of the trapezoid in this problem is given as follows:

A = 27a³b² square units.

The area of a trapezoid is given by half the multiplication of the height by the sum of the bases, hence:

A = 0.5 x h x (b1 + b2).

The parameters for this problem are given as follows:

h = 3ab, b1 = 7a²b, b2 = 11a²b.

Hence the area is given as follows:

A = 0.5 x 3ab x 18a²b.

A = 27a³b² square units.

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The right response is D) Because the matching sides of the triangles are not proportional, AABC and AFED are not similar.

We must compare the ratios of the respective sides of the triangles AABC and AFED in order to determine whether they are similar.

Given:

AABC: AC = 12, AB = 4, BC = 6

AFED: DF = 15, DE = 7.5, EF = 5

The respective side ratios must be the same for triangles to be considered comparable. We'll figure out the ratios for the appropriate sides as follows:

Ratio of AC to DF: AC/DF = 12/15 = 0.8

Ratio of AB to DE: AB/DE = 4/7.5 = 0.533

Ratio of BC to EF: BC/EF = 6/5 = 1.2

The triangles AABC and AFED are not comparable because the ratios of the corresponding sides are not equal.

Because the matching sides of the triangles are not proportional, the correct response is D) AABC and AFED are not similar. The ratios of the matching sides do not line up, proving that there is no similarity between the triangles.

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The probability that a randomly selected point within the square falls in the red shaded square is 1/16

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event to occur is 1 and it is 100% in percentage.

Probability = sample space /total outcome

sample space is the area of the red shaded square and the total outcome is the big square.

Area of red shaded square = 1 × 1 = 1unit²

area of the big square = 4 × 4 = 16 units²

Therefore the probability that a point selected falls on the red shaded square

= 1/16

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The surface area of Seung's block is approximately 3 times greater than Derek's block , option (B).

To compare the surface areas of Seung's and Derek's wooden blocks, we need to calculate the surface areas of both blocks and find the ratio between them.

Seung's block:

The surface area of a square pyramid can be calculated using the formula:

Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height

Given that the height of Seung's block is 6 cm and the base is 10 cm, we can calculate the slant height using the Pythagorean theorem:

Slant Height = sqrt((1/2 * base)^2 + height^2)

            = sqrt((1/2 * 10 cm)^2 + (6 cm)^2)

            = sqrt(25 cm^2 + 36 cm^2)

            = sqrt(61 cm^2)

            ≈ 7.81 cm

Now we can calculate the surface area of Seung's block:

Surface Area of Seung's Block = 10 cm^2 + (1/2 * 10 cm * 7.81 cm)

                            = 10 cm^2 + 39.05 cm^2

                            = 49.05 cm^2

Derek's block:

Since Derek's block has dimensions that are half of Seung's block, the base of Derek's block will be 10 cm / 2 = 5 cm, and the height will be 6 cm / 2 = 3 cm.

The surface area of Derek's block can be calculated using the same formula:

Surface Area of Derek's Block = 5 cm^2 + (1/2 * 5 cm * slant height of Derek's block)

To find the slant height of Derek's block, we can use a similar approach as before:

Slant Height of Derek's Block = sqrt((1/2 * 5 cm)^2 + (3 cm)^2)

                            = sqrt(6.25 cm^2 + 9 cm^2)

                            = sqrt(15.25 cm^2)

                            ≈ 3.90 cm

Now we can calculate the surface area of Derek's block:

Surface Area of Derek's Block = 5 cm^2 + (1/2 * 5 cm * 3.90 cm)

                            = 5 cm^2 + 9.75 cm^2

                            = 14.75 cm^2

Finally, we can find the ratio of the surface area of Seung's block to Derek's block:

Surface Area Ratio = Surface Area of Seung's Block / Surface Area of Derek's Block

                 = 49.05 cm^2 / 14.75 cm^2

                 ≈ 3.32

Rounded to the nearest whole number, the surface area of Seung's block is approximately 3 times greater than Derek's block. Therefore, the answer is option B: 3.

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Using the cross product for the proff of Pythagoras theorem, the correct step is

By the cross product property, AB² =  BC multiplied by BD

The cross product property is typically used to solve equations with two values, where the product of the extremes (the outer terms) is equal to the product of the means (the inner terms).

For the similar triangles, the ratio is as follows

BD / BA = BA / BC

BA² = BD * BC

and AB = BA, hence

BA² = BD * BC

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To sketch the graph of the function f(x) = 2cos^2(x) - 3x + 4 for the values of x between 0 and 2, we can follow these steps:

1. Determine key points: Find the critical points and important features of the function within the given range. This includes the x-intercepts, local maxima, and minima.

2. Plot the x-intercepts: To find the x-intercepts, set f(x) = 0 and solve for x. In this case, cos^2(x) = (0 + 3x - 4)/2. Find the values of x where this equation holds true.

3. Identify local maxima and minima: Use calculus or other methods to find the local maximum and minimum points of the function within the given range.

4. Sketch the graph: Once you have the key points, plot them on a graph and connect them smoothly. Pay attention to the shape and behavior of the function.

Since the process of sketching a graph is best done visually, I'm unable to provide an image here. However, by following the steps outlined above, you can plot the graph of f(x) within the range of x between 0 and 2. Remember to label the axes and indicate any important points you have found.

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Answer:

No

Step-by-step explanation:

To determine whether the given set of ordered pairs {(49,13),(61,36),(10,27),(76,52),(23,52)} represents a function, check if each x-value is associated with a unique y-value.

Check the x-values in the set: 49, 61, 10, 76, and 23. There are no repeated x-values.  Still need to check if each x-value has a unique corresponding y-value.

Check the y-values: 13, 36, 27, 52, and 52. There is one repeated y-value, 52, for the pairs (76, 52) and (23, 52).

Conclusion: the y-value 52 is associated with 2 different x-values, therefore the given set of ordered pairs does not represent a function.

Answer: To find the total distance Reena covers in 6 days of a week, we need to consider the distance between her school and house for a single round trip and then multiply it by 6.

Given that the distance between the school and Reena's house is 1 km 480 m, we can add the distance for a one-way trip to get the total round trip distance.

Total round trip distance = 1 km 480 m + 1 km 480 m = 2 km 960 m

Now, to find the distance covered by Reena in 6 days, we multiply the round trip distance by 6:

Total distance covered in 6 days = 2 km 960 m * 6

Converting the distance to meters for easier calculation:

2 km 960 m = 2960 m

Total distance covered in 6 days = 2960 m * 6

Calculating the total distance covered:

Total distance covered in 6 days = 17760 m

Therefore, Reena covers a total distance of 17,760 meters (or 17.76 kilometers) in 6 days of a week.

The length of the arc is approximately 61.68 cm, and the area of the sector is approximately 1509.16 cm^2.

To find the length of the arc and the area of the sector, we can use the formulas related to circles and angles.

Length of the arc:

The formula to find the length of an arc in a circle is given by:

Arc Length = (angle / 360°) × (2πr)

In this case, the angle is 72°, and the radius is 49 cm. Substituting these values into the formula, we get:

Arc Length = (72° / 360°) × (2π × 49 cm)

Arc Length = (1/5) × (2π × 49 cm)

Arc Length = (2/5) × (π × 49 cm)

Arc Length = (2/5) × (π × 49 cm)

Arc Length ≈ 61.68 cm (rounded to the nearest hundredth)

Therefore, the length of the arc is approximately 61.68 cm.

Area of the sector:

To calculate the area of the sector, we need to use the formula:

Sector Area = (θ / 360°) × (π × r^2)

where θ is the central angle in degrees and r is the radius of the sector.

In this case, the central angle is 72° and the radius is 49 cm. Plugging these values into the formula, we have:

Sector Area = (72° / 360°) × (π × (49 cm)^2)

= (1/5) × (π × 2401 cm^2)

≈ (1/5) × (3.14159 × 2401 cm^2)

≈ (1/5) × 7545.8 cm^2

≈ 1509.16 cm^2

Therefore, the area of the sector is approximately 1509.16 cm^2.

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The independent variable in this data is the speed of the car. This is because the congressman is trying to determine whether or not the gas mileage improves at lower speeds.

The dependent variable is the gas mileage, because it is the variable that is being measured and that is affected by the independent variable.

The other variables in this data are the weight of the car, the type of engine, and the year of the car. However, these variables are not being changed in this experiment, so they are not considered to be independent variables.

The data shows that the gas mileage of the Toyota Camry improves as the speed of the car decreases. This supports the claim of the supporters of the bill to reduce the speed limit.

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Answer

a. The shape of the cross section is a rectangle.

b. To find the perimeter of the cross section, we need to determine the length of its sides.

In a cube with an edge length of 6 inches, when a plane intersects diagonally, it creates a rectangle with dimensions of the cube's diagonal and half of its edge length.

The diagonal of the cube can be found using the Pythagorean theorem:

diagonal = √(edge length^2 + edge length^2 + edge length^2) = √(6^2 + 6^2 + 6^2) = √(36 + 36 + 36) = √108 ≈ 10.39 inches

The length and width of the rectangle are half of the edge length, which is 6/2 = 3 inches.

Since the cross section is a rectangle, its perimeter is given by:

perimeter = 2 * (length + width) = 2 * (3 + 10.39) ≈ 26.78 inches

Therefore, the perimeter of the cross section, to the nearest hundredth, is approximately 26.78 inches.

c. To find the area of the cross section, we multiply the length and width of the rectangle:

area = length * width = 3 * 10.39 ≈ 31.17 square inches

Therefore, the area of the cross section, to the nearest hundredth, is approximately 31.17 square inches.

Step-by-step explanation: