Find the distance between the following pairs of points.

(2,-1),(7,-1)

The exact values of trigonometric functions are, respectively:

sin (u + v) = - 13 /85

tan (u + v) = - 13 / 84

In this problem we need to determine the exact values of trigonometric functions, this can be done by using trigonometric formulas and relationships between trigonometric functions. We need to use the following expressions:

sin² x + cos² x = 1

sin (u + v) = sin u · cos v + cos u · sin v

tan x = sin x / cos x

tan (u + v) = (tan u + tan v) / (1 - tan u · tan v)

Where x, u, v are measured in radians.

Now we proceed to determine the exact values of each function:

cos u = √[1 - (- 3 / 5)²]

cos u = 4 / 5

sin v = √[1 - (15 / 17)²]

sin v = 8 / 17

sin (u + v) = (- 3 / 5) · (15 / 17) + (4 / 5) · (8 / 17)

sin (u + v) = - 13 /85

tan u = (- 3 / 5) / (4 / 5)

tan u = - 3 / 4

tan v = (8 / 17) / (15 / 17)

tan v = 8 / 15

tan (u + v) = (- 3 / 4 + 8 / 15) / [1 - (- 3 / 4) · (8 / 15)]

tan (u + v) = - 13 / 84

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Answer:

x = 2;

y = √3

Step-by-step explanation:

Use trigonometry:

[tex] \tan(60°) = \frac{y}{1} [/tex]

Cross-multiply to find y:

[tex]y = 1 \times \tan(60°) = 1 \times \sqrt{3} = \sqrt{3} [/tex]

Use the Pythagorean theorem to find x:

[tex] {x}^{2} = {y}^{2} + {1}^{2} [/tex]

[tex] {x}^{2} = ( { \sqrt{3}) }^{2} + {1}^{2} = 3 + 1 = 4[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{4} = 2[/tex]

The simplified form of the function f(x)=1/2(27) 2x/3 can be expressed as f(x) = (27/2) (2x/3).

Functions are one of the fundamental building blocks of mathematics and are used to describe and analyze relationships between different variables.

The function f(x)=1/2(27) 2x/3 can be simplified by factoring out the common factor of 1/2 and 27.

Thus, the simplified form of the function can be expressed as

f(x) = (27/2) (2x/3).

This function is a polynomial function with degree 1, which means that it is a linear function. The degree of a function is the highest power of the variable in the equation.

The key aspects of this function can be identified by looking at the constant values in the equation.

The constant value 27/2 is the y-intercept, which is the point at which the line crosses the y-axis.

This means that the y-value of the function at x = 0 is 27/2.

The constant value 2/3 is the gradient, which is the slope of the line. This means that for every increase in the x-value, the y-value will increase by 2/3.

This function can be represented graphically as a straight line with a y-intercept of 27/2 and a slope of 2/3.

The graph of this function will pass through the point (0, 27/2) and will have a positive slope of 2/3. This means that the graph will move up and to the right, with each increase in the x-value resulting in an increase of 2/3 in the y-value.

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Using dilation, we can find the coordinates of the new trapezoid and the coordinates of T" is (-6, -4).

Dilation is the process of increasing an object's size without altering its shape. Depending on the scale-factor, the object's size may increase or shrink. A square of side 5 units can be widened to a square of side 15 units using dilation maths, but the square's shape doesn't change.

In geometry, dilation math is used to enlarge and reduce two- or three-dimensional figures.

Here in the question,

The coordinates of point T = (-3, -2)

Now, the trapezoid is dilated about the origin.

The scale factor here is. k = 2.

The new coordinates of the point T":

x coordinate = -3 × 2 = -6

y coordinate= -2 × 2 = -4

The coordinates of T" = (-6, -4).

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the approximate annual cost of raising a child born in 2013 at age 17 is about $15,050, assuming that the trend of increasing cost per year continues. we first need to calculate the slope (m) and the y-intercept (b)

what do you mean by approximate ?

Approximate means "close to but not exactly" or "an estimate". In other words, an approximate value is a value that is not exact but is close enough to the actual value to be useful or meaningful in a given context.

In the given question,

To find the equation of the line of fit in slope-intercept form, we first need to calculate the slope (m) and the y-intercept (b) using the two given points (8, 12,800) and (14, 14,420).

m = (y₂ - y₁) / (x₂ - x₁)

m = (14,420 - 12,800) / (14 - 8)

m = 1,620 / 6

m = 270

b = y - mx

b = 12,800 - 270(8)

b = 10,940

Therefore, the equation of the line of fit in slope-intercept form is:

y = mx + b

y = 270x + 10,940

To use this equation to estimate the cost of raising a child born in 2013 at age 17, we can plug in x = 17 and solve for y:

y = 270x + 10,940

y = 270(17) + 10,940

y = 15,050

Therefore, the approximate annual cost of raising a child born in 2013 at age 17 is about $15,050, assuming that the trend of increasing cost per year continues.

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Answer:5

Step-by-step explanation:

Answer:

Step-by-step explanation:Let's first substitute the values of A, P, and T in the formula A = P(1 + r)^T.

Given: A = 1,000,000; P = 10,000; T = 2

1,000,000 = 10,000(1 + r)^2

Now let's express it as a quadratic equation. First, divide both sides by 10,000:

100 = (1 + r)^2

Next, expand the square:

100 = 1 + 2r + r^2

Finally, rearrange to form the quadratic equation:

r^2 + 2r - 99 = 0

To represent the formula A = P(1 + r) as a quadratic equation with the given values, we substitute A = 1,000,000, P = 10,000, and T = 2. The resulting equation is 10,000r - 9,000,000 = 0.

To express the formula A = P(1 + r) as a quadratic equation, we substitute the given values for A, P, and T.

Given:

Substituting these values into the formula:

1,000,000 = 10,000(1 + r)

To express this equation as a quadratic form, we can expand the parentheses:

1,000,000 = 10,000 + 10,000r

Now, let's move all the terms to one side to form a quadratic equation:

10,000r - 9,000,000 = 0

Therefore, the quadratic equation that represents A = P(1 + r) with the given values is 10,000r - 9,000,000 = 0.

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Answer:

It took Adele's brother 37.45 + 2.10 = 39.55 seconds to run the race.

Answer: We can determine which graph represents which polynomial function by analyzing the factors of each function.

First, let's consider f(x) = (x + 3)(x + 4). The factors are (x + 3) and (x + 4). When we multiply these factors together, we get a quadratic polynomial with a positive leading coefficient. This means that the graph of f(x) will be a parabola that opens upward.

Next, let's consider g(x) = 1/3(x + 3)(x − 4). The factors are (x + 3) and (x - 4). When we multiply these factors together and simplify, we get a quadratic polynomial with a leading coefficient of 1/3. This means that the graph of g(x) will also be a parabola that opens upward, but it will be narrower than the graph of f(x).

Based on this analysis, we can determine that the graph of f(x) corresponds to the wider parabola, and the graph of g(x) corresponds to the narrower parabola. We can also determine this without using graphing software by noting that f(x) has roots at x = -3 and x = -4, while g(x) has roots at x = -3 and x = 4. The graph of f(x) must therefore intersect the x-axis at -3 and -4, while the graph of g(x) must intersect the x-axis at -3 and 4. By examining the graphs, we can see that the wider parabola intersects the x-axis at -3 and -4, so it corresponds to f(x), while the narrower parabola intersects the x-axis at -3 and 4, so it corresponds to g(x).

Step-by-step explanation:

To compute the CPI, we first need to calculate the total cost of the basket in both years:

2017: (10 x $40) + (30 x $10) = $400 + $300 = $700

2018: (12 x $60) + (50 x $12) = $720 + $600 = $1320

Using 2017 as the base year, the CPI in 2017 is 100 (by definition). To calculate the CPI in 2018, we divide the cost of the basket in 2018 by the cost of the basket in 2017, and multiply by 100:

CPI in 2018 = (1320/700) x 100 = 188.57

Therefore, the percentage change in the overall price level is:

% change in price level = (CPI in 2018 - CPI in 2017) / CPI in 2017 x 100

% change in price level = (188.57 - 100) / 100 x 100

% change in price level = 88.57%

So, the overall price level increased by 88.57% from 2017 to 2018.

If one number is increased by a factor of 12 and the other is decreased by a factor of 4, the product of the two numbers will be multiplied by a factor of 3.

Let's suppose we have two numbers, A and B, and we want to know how their product will change if one number is increased by a factor of 12 and the other is decreased by a factor of 4.

The initial product of the two numbers is:

A x B

If we increase A by a factor of 12, the new value of A will be 12A. If we decrease B by a factor of 4, the new value of B will be B/4. Therefore, the new product of the two numbers will be:

(12A) x (B/4) = (12/4) x A x B = 3AB

So the new product of the two numbers will be three times the initial product. In other words, if one number is increased by a factor of 12 and the other is decreased by a factor of 4, the product of the two numbers will increase by a factor of 3.

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[ 25 , 1.581 ] is the pair that is the mean and standard error of x.

What does standard error mean?

A statistical concept known as the standard error uses standard deviation to assess how well a sample distribution represents a population.

                            The standard error of the mean describes the statistical variation between a sample mean and the population's actual mean. Measures of variability include standard error and standard deviation: The standard deviation describes variation within a single sample.

 n = 80

 μ = 25

 σ² = 200

mean of x =  μ  = 25

 standard error = √ σ²/n

                        = √200/80

                         = 1.581

                         = [ 25 , 1.581 ]

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Answer:

Food is brought for us by them.

Answer:

Food is brought for us by them.

Step-by-step explanation:

Answer:

x = 25

Step-by-step explanation:

By the angle sum property,

2x + 2 + 5x + 3 = 180

7x + 5 = 180

7x = 175

x = 25

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Answer:

x = 25

Step-by-step explanation:

2x + 2 + 5x + 3 = 180

7x + 5 = 180 and 7x=175

7x = 175

x = 25

so x is most  likely the answer

x=25 Give her BRAINLEST for figuring it out first

For the four transformations based on f(x)= [tex]2^{x}[/tex],

1)6f(x)= 6 [tex]2^{x}[/tex]

2)f(6x)= [tex]2^{6x}[/tex]

3)f(x+6)=[tex]2^{x+6}[/tex]

4)f(x)+6 =  [tex]2^{x}[/tex]+6

What are transformations?

Transformations in any given function is changing its original form to nre form by flipping, rotating, shifting, enlarging and compressing the function. We can move the given function up or down as per the given conditions by adding up or subtracting the constant in y axis. We can move the given function left or right as per the given conditions by adding up or subtracting the constant in x axis. We can stretch or compress the function about x or y axes. Also we can also flip,reverse the function, reflect about axes or enlarge the functions.

Here given that function f(x)= [tex]2^{x}[/tex]

From the given graph we can identify few points for the given function:

(-1,0.5); (0,1); (1,2); (2,4); (3,8); (4,16); (5,32); (6,64) and so on.

Now to identify the transformations, we can substitute the tranformed value of x in the function:

1)6 f(x) = 6 .  [tex]2^{x}[/tex] {as we know that f(x)= [tex]2^{x}[/tex]}

∴6 f(x) will be equal to 6 .  [tex]2^{x}[/tex]

2)f(6x) : for this we can replace 'x' by '6x'

f(6x) =  [tex]2^{6x}[/tex]

3)f(x+6): for this function replace 'x' by 'x+6'

f(x+6)=[tex]2^{x+6}[/tex]

4)f(x)+6 : substitute f(x)= [tex]2^{x}[/tex], we get

f(x)+6 =  [tex]2^{x}[/tex]+6

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Answer:

7 days

Step-by-step explanation:

The slope of a line perpendicular to [tex]2x + 3y = 24[/tex] is 3/2

An equation is a statement that two expressions are equal. It contains one or more variables and may also contain constants, coefficients, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.

To find the slope of a line perpendicular to the line whose equation is [tex]2x + 3y = 24[/tex], we need to first find the slope of the given line.

We can rearrange the equation into slope-intercept form (y = mx + b) by solving for y:

[tex]2x + 3y = 24[/tex]

[tex]3y = -2x + 24[/tex]

[tex]y = \huge \text(-\dfrac{2}{3}\huge \text )x + 8[/tex]

So the slope of the given line is -2/3.

A line perpendicular to this line will have a slope that is the negative reciprocal of -2/3.

The negative reciprocal of a number is the number flipped upside down and then negated. So the negative reciprocal of -2/3 is:

[tex]-1 \div \huge \text(-\dfrac{2}{3}\huge \text )=\dfrac{3}{2}[/tex]

Therefore, the slope of a line perpendicular to [tex]2x + 3y = 24[/tex] is 3/2

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Answer:

Perimeter: Add up all the sides (7.5, 6, 2.5, 3.5, 3.5) to get the answer for the perimeter as 23 ft.

Area: Split the shape into two rectangles either way and multiply the length and width. Then, add the two answers. This would leave us with the answer of the area as 35 square feet.

Answer:

[tex]\tan \left(\dfrac{x}{2}\right)=3[/tex]

Step-by-step explanation:

Trigonometric ratios are the ratios of the sides of a right triangle.

[tex]\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]

The sine trigonometric ratio is the ratio of the side opposite the angle to the hypotenuse.

Given sin(x) = 3/5, the side opposite angle x is 3, and the hypotenuse is 5.

As we have two sides of the right triangle, we can calculate the third side (the side adjacent the angle) using Pythagoras Theorem.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

Therefore:

[tex]\implies A^2+3^2=5^2[/tex]

[tex]\implies A^2+9=25[/tex]

[tex]\implies A^2+9-9=25-9[/tex]

[tex]\implies A^2=16[/tex]

[tex]\implies \sqrt{A^2}=\sqrt{16}[/tex]

[tex]\implies A=4[/tex]

Use the cosine trigonometric ratio to find the value of cos(x), remembering that cosine is negative in Quadrant II.

[tex]\implies \cos x=-\dfrac{4}{5}[/tex]

Now we have the values of sin(x) and cos(x) in Quadrant II, we can use the tangent half angle formula to find the value of tan(x/2).

[tex]\begin{aligned}\implies \tan \left(\dfrac{x}{2}\right)&=\dfrac{\sin x}{1+\cos x}\\\\&=\dfrac{\frac{3}{5}}{1-\frac{4}{5}}\\\\&=\dfrac{\frac{3}{5}}{\frac{1}{5}}\\\\&=\dfrac{3}{5} \cdot \frac{5}{1}\\\\&=3\end{aligned}[/tex]

Therefore, the value of tan(x/2) is 3.

Answer: The height above the ground in feet of a football thrown into the air from the balcony of a house is given by the expression:

h(t) = -12t^2 + 20t + 30

where t is the time in seconds since the ball was thrown.

To find the height of the balcony above the ground, we need to determine the initial height of the football when it was thrown from the balcony. This initial height corresponds to the value of h(0), since the time elapsed since the throw was zero at that moment.

Therefore, we can substitute t = 0 into the expression for h(t):

h(0) = -12(0)^2 + 20(0) + 30 = 30

This means that the balcony is 30 feet above the ground.

Hence, the height of the balcony above the ground is 30 feet.

Step-by-step explanation:

The length of the parking lot in the drawing is 44 centimeters.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and operators (such as addition, subtraction, multiplication, and division) that are used to represent quantities and their relationships.

Since the scale Colton used is 1 centimeter : 2 meters, this means that for every 2 meters in real life, he drew 1 centimeter in his scale drawing.

To find the length of the parking lot in the drawing, we can set up a proportion:

1 cm / 2 m = x cm / 88 m

where x is the length of the parking lot in the drawing in centimeters.

We can solve for x by cross-multiplying:

1 cm * 88 m = 2 m * x cm

88 cm = 2x

x = 44 cm

Therefore, the length of the parking lot in the drawing is 44 centimeters.

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The correct option is $58,527.87

Using the loan details and bonus payment provided, the remaining balance on the loan at the end of 10 years and a month can be calculated as follows:

Number of payments made = 10 years * 12 months/year + 1 month = 121

Monthly interest rate = 4.75% / 12 = 0.3958%

Yearly bonus payment = $4,500

Using the PMT function in Excel, the monthly payment on the loan can be calculated as:

PMT(0.003958, 15*12, 320000) = -$2,378.03

Since the bonus payment is made once a year, it can be applied as a lump sum to the remaining balance at the end of each year. Therefore, the remaining balance after 10 years and a month can be calculated as:

Remaining balance = PV(0.003958, 5*12, -2378.03, 0, 0) - 4500

where PV is the present value function.

Solving this equation yields:

Remaining balance = $58,527.87

Therefore, the answer is $58,527.87.

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sandhyamahilane1116

24.02.2021

Math

Secondary School

answered • expert verified

What is a fraction between 6/7 and 1 whole?

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Given : 6/7 and 1 whole

To Find : a fraction between 6/7 and 1 whole

Solution:

There exist Infinitely many rational between two different rational numbers

There can be Different ways to find rational number in between two number

one way is to find mean of number

= (6/7 + 1)/2

= 13/14

13/14 is a fraction between 6/7 and 1 whole

6/7 = 18/21

1 = 21/21

19/21 , 20/21 are fraction between 6/7 andfraction

The value of k in the expression x³ – 6x² – 11x + k is –10.

Polynomial is formed up of the terms Nominal, which means "terms," and Poly, which means "many." A polynomial is a mathematical expression made up of variables, constants, and exponents that are mixed using addition, subtraction, multiplication, and division operations.

From the question given above, the following data were obtained:

f(x) = x³ – 6x² – 11x + k

Factor => x + 2

Value of k =?

Next, we shall obtained the value of x from x + 2. This can be obtained as follow:

x + 2 = 0

Collect like terms

x = 0 – 2

x = –2

Finally, we shall determine the the value of k. This can be obtained as  illustrated below:

f(x) = x³ – 6x² – 11x + k

x = –2

Value of k =?  

f(–2) = 0 since x + 2 is a factor

x³ – 6x² – 11x + k = 0

(–2)³ – (–2)² – 11(–2) + k = 0

–8 – (4) + 22 + K = 0

–8 – 4 + 22 + K = 0

10 + k = 0

Collect like terms

k = 0 – 10

k = –10

Therefore, the value of k is –10

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Answer:

y = 2;

x = 2√2

.

Step-by-step explanation:

Use trigonometry:

[tex] \tan(45°) = \frac{y}{2} [/tex]

Use the property of proportion to find x (cross-multiply):

[tex]y = 2 \times \tan(45°) = 2 \times 1 = 2[/tex]

Use the Pythagorean theorem to find x:

[tex] {x}^{2} = {y}^{2} + {2}^{2} [/tex]

[tex] {x}^{2} = {2}^{2} + {2}^{2} = 4 + 4 = 8[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{8} = \sqrt{4 \times 2} = 2 \sqrt{2} [/tex]

Answer:

To determine the area of the garden that needs to be covered with plastic, we need to multiply the length by the width of the garden. However, we need to convert the measurements to the same unit of measurement. Let's convert the measurements into yards since we need to find the area in square yards. 12 ft = 4 yards 15 ft = 5 yards 9 ft = 3 yards Now, we can calculate the area: Area = Length x Width Area = 5 yards x 4 yards Area = 20 square yards Therefore, Robert needs 20 square yards of plastic to cover his garden.

Answer:

x = 4

Step-by-step explanation:

In order to find x, we will either have to find the length QT or RT.

In triangle RST, the length RS is given so we can either length RT or ST. However, if we find the length RT, we can find the value of x.

sin Θ = [tex]\frac{opp}{hyp}[/tex]

sin 60° = [tex]\frac{2\sqrt{3} }{RT}[/tex]

RT = [tex]\frac{2\sqrt{3}}{sin 60}[/tex] = 4

tan Θ = [tex]\frac{opp}{adj}[/tex]

tan 45° = [tex]\frac{x}{4}[/tex]

x = tan 45° × 4 = 4

The indicated side lengths are: x ≈ 13.2 units and y ≈ 11.9 units

the ratio between the side opposite the angle and the hypotenuse of the triangle.

given that a triangle with three sides x, y, 18 and angle opposite to side y is 42°.

then by sine formula, trigonometric ratio for the sine of an angle:

sin(42°) = perpendicular /hypotenuse = y/18

y = 18 sin(42°), we find that y ≈ 11.9 units (rounded to the nearest tenth).

To find x,  the cosine of an angle:

cos(42°) =base /hypotenuse=x/18

x = 18 cos(42°)

we find that x ≈ 13.2 units (rounded to the nearest tenth).

Therefore, the indicated side lengths are:

x ≈ 13.2 units

y ≈ 11.9 units

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a)The range of 95% of the data is from 21 mpg to 26 mpg, which is a range of 5 mpg.

b)We need a sample size of 543 to estimate the mean with 98% confidence and an error of 0.25 mpg

What is Empirical Rule for a normal distribution?

If a dataset is normally distributed, we can expect that about 68% of the data points will fall within one standard deviation of the mean, about 95% of the data points will fall within two standard deviations of the mean, and about 99.7% of the data points will fall within three standard deviations of the mean. This rule is a useful guideline for understanding the spread of data in a normal distribution.

(a) Using Method 3 (the Empirical Rule for a normal distribution), we know that for a normally distributed random variable, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean.

Since the range of the combined city/highway fuel economy of a 2016 Toyota 4Runner 2WD 6-cylinder 4-L automatic 5-speed using regular gas is from 21 mpg to 26 mpg, the midpoint of the range is (21 + 26) / 2 = 23.5 mpg.

Using the Empirical rule, we know that approximately 95% of the data falls within two standard deviations of the mean. Therefore, the range of 95% of the data is from 21 mpg to 26 mpg, which is a range of 5 mpg.

We can set up the following equation to solve for the standard deviation, σ:

2σ = 5

σ = 5 / 2

σ = 2.5

Therefore, the estimated standard deviation is 2.5 mpg. Rounded to 4 decimal places, the estimated standard deviation is 2.5000 mpg.

(b) The formula for the margin of error is:

Margin of error = z-value×(standard deviation / √(sample size))

We want the margin of error to be 0.25 mpg and the confidence level to be 98%. Since we are using a z-value, we can look up the z-value for a 98% confidence level in a standard normal distribution table.

The z-value for a 98% confidence level is approximately 2.33 when rounded to 3 decimal places.

Plugging in the given values, we have:

0.25 = 2.33×(2.5 / √(sample size))

Solving for the sample size, we get:

√(sample size) = 2.33 × (2.5 / 0.25)

√(sample size) = 23.3

sample size = (23.3)²

sample size = 542.89

Rounded to the nearest whole number, we need a sample size of 543 to estimate the mean with 98% confidence and an error of 0.25 mpg.

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