Consider the function f(x) = (x - 3)^2
a. How could you restrict the domain of f(x) so that its inverse will be a function?
b. Graph f(x) with its restricted domain and then graph its inverse on the same set of axes.
c. Find the equation of the inverse of f(x) with its restricted domain.

Answer:

k is 2.5.

Step-by-step explanation:

To find the value of k, we can use the information given about points A and B to find the value of the parameters a and b. Since we know that the curve passes through A (1, 10) and B (4, 80), we can substitute these points into the equation y = ab^x to find the values of a and b.

Starting with point A:

10 = a * b^1

Next, using point B:

80 = a * b^4

We can now use these two equations to find the value of b:

10 = a * b^1

80 = a * b^4

Dividing the second equation by the first equation:

8 = b^3

Taking the cube root of both sides:

b = 2

We can now use either of the two equations to find the value of a:

10 = a * b^1

a = 10/2 = 5

Now that we have found the values of a and b, we can substitute the point C (-1, k) into the equation y = ab^x to find the value of k:

k = 5 * 2^(-1) = 5 * (1/2) = 2.5

So the value of k is 2.5.

Answer: b1 = [tex]\frac{2A}{h} -b2[/tex]

Step-by-step explanation:

Algebra my friend...Algebra

So:

A = 0.5h(b1+b2)

A/(0.5h) =b1+b2

.: A/(0.5h) - b2 = b1

In other words:

b1 = [tex]\frac{A}{0.5h} -b2[/tex]

The 0.5 in the denominator can be rearranged to get the final answer:

b1 = [tex]\frac{2A}{h} -b2[/tex]

The probability that between 19 and 53 (including 19 and 53) seeds grow into healthy plants is 0.9941.

The probability distribution of x, where x is the number of seeds that grow into healthy plants when 75 seeds are planted, is a binomial distribution with parameters n = 75 and p = 0.69. The probability of x = 44 is calculated by using the formula for the probability mass function of the binomial distribution:

[tex]P(x=44) = (75 choose 44) * (0.69)^44 * (1-0.69)^(75-44) = 0.1578[/tex]

The probability of x >= 44 is calculated by using the formula for the cumulative probability function of the binomial distribution:

[tex]P(x > = 44) = 1 - P(x < = 43) = 1 - (75 choose 43) * (0.69)^43 * (1-0.69)^(75-43) = 0.7406[/tex]

The probability of x > 44 is calculated by subtracting the probability of x = 44 from the probability of x >= 44:

P(x > 44) = P(x >= 44) - P(x = 44) = 0.7406 - 0.1578 = 0.5828

The probability of 19 <= x <= 53 is calculated by subtracting the probability of x <= 18 from the probability of x <= 53:

[tex]P(19 < =x < =53) = P(x < =53) - P(x < =18) = 0.9994 - 0.0053 = 0.9941[/tex]

Therefore, the probability that exactly 44 seeds grow into healthy plants is 0.1578, the probability that at least 44 seeds grow into healthy plants is 0.7406, the probability that more than 44 seeds grow into healthy plants is 0.5828, and the probability that between 19 and 53 (including 19 and 53) seeds grow into healthy plants is 0.9941.

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The volume of the sand which is equal to the volume of the cone will be 376.8 cubic feet.

Let h be the height of the cone and A be the base area of the cone.

Then the volume of the cone will be given as,

Volume = (1/3) × A × h

At a construction site, sand is poured from a truck into a conical pile. The pile has a diameter of 12 feet and is 10 feet tall.

The base area of the cone is given as,

A = π × (d/2)²

A = 3.14 × (12 / 2)²

A = 3.14 × 36

A = 113.04 square feet

Then the volume of the cone is given as,

V = 1/3 × 113.04 × 10

V = 376.8 cubic feet

The volume of the sand which is equal to the volume of the cone will be 376.8 cubic feet.

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Answer: 96

The sum of the interior angles of a regular heptagon is equal to (n-2)180 degrees, where n is the number of sides of the polygon.

Since a regular heptagon has 7 sides, the sum of its interior angles is (7-2)180 = 720 degrees.

Therefore, the measure of each interior angle is 720 degrees / 7 = 104 degrees.

If the measure of an interior angle is (w + 8) degrees, then w + 8 = 104.

Solving for w, we get w = 96.

The required perimeter of the door on the scale drawing is 4 cm, which is option A.

Perimeter is the measure of the figure on its circumference.

Here,
To find the perimeter of the door on the scale drawing, we need to first find the dimensions of the door on the scale drawing. We can use the scale 2:8 to convert the actual dimensions of the door to the corresponding dimensions on the scale drawing:

Height on scale drawing = (2/8) x 5 cm = 1.25 cm

Width on scale drawing = (2/8) x 3 cm = 0.75 cm

The perimeter on the scale drawing = 2 x (height on scale drawing + width on scale drawing)

Perimeter on scale drawing = 2 x (1.25 cm + 0.75 cm)

Perimeter on scale drawing = 2 x 2 cm

The perimeter on the scale drawing = 4 cm

Therefore, the perimeter of the door on the scale drawing is 4 cm, which is option A.

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For given circle, Area of shaded region is 52π cm².

What exactly is a circle?

A circle is a kind of ellipse with zero eccentricity and two foci that are coincident. A circle is also known as the locus of points drawn at equal distances from the center. The radius of a circle is the distance from its center to its outside line. The diameter of a circle is the line that divides it into two equal sections and is equal to twice the radius.

The equation for a circle in the plane is:

(x-h)^²+ (y-k)² = r²

When the coordinate points are (x, y)

(h, k) is the coordinate of a circle's center.

where r is the circumference of a circle.

where circle area = πr²

Circle circumference=2πr

Now,

Radius of biggest circle = 8cm

radius of unshaded circle= 4cm and

radius of smaller shaded circle= 2cm

Then,

Area of shaded region= Area of biggest circle - area of white circle + area of smaller shaded circle

=π*8² - π*4² + π*2²

=64π-16π+4π

=52π cm²

hence,

           Area of shaded region is 52π cm².

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Using the gradient of the curve given, the value of a is 6

The stationary point of a curve is a point where the slope of the curve is equal to zero. So, to find the value of a, we need to set the derivative of the curve equal to zero and solve for x.

dy/dx = 2(x + 3)^(1/2) - x

Setting this equal to zero, we have:

2(x + 3)^(1/2) - x = 0

Expanding the square root and rearranging, we get:

x = 2(x + 3)^(1/2)

Squaring both sides of the equation, we have:

x^2 = 4(x + 3)

Expanding the right side and rearranging, we have:

x^2 - 4x - 12 = 0

Using the quadratic formula, we can find the values of x that satisfy this equation:

x = [-(-4) ± √((-4)^2 - 4(1)(-12))] / 2(1)

x = [4 ± √(16 + 48)] / 2

x = [4 ± √64] / 2

x = [4 ± 8] / 2

So, the two possible values of x are:

x = 6, x = -2

Since we are looking for a positive value of x, the only solution that works is x = 6.

Therefore, the value of a is equal to 6.

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The proportion of the time that Mrs. Starnes finishes an easy sudoku puzzle in less than 3 minutes is approximately 0.0055, or 0.55%

We can use the normal distribution and the given mean and standard deviation to find the proportion of the time that Mrs. Starnes finishes an easy sudoku puzzle in less than 3 minutes.

First, we need to standardize the value of 3 minutes using the z-score formula:

z = (x - μ) / σ

where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

Substituting the given values, we get:

z = (3 - 5.3) / 0.9 ≈ -2.56

Next, we can use a standard normal distribution table or a calculator to find the proportion of the area under the standard normal distribution curve that corresponds to a z-score of -2.56. This proportion represents the proportion of the time that Mrs. Starnes finishes an easy sudoku puzzle in less than 3 minutes.

Using a standard normal distribution table, we can find that the area to the left of z = -2.56 is approximately 0.0055.

Therefore,

The proportion of the time that Mrs. Starnes finishes an easy sudoku puzzle in less than 3 minutes is approximately 0.0055, or 0.55%.

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Using the empirical rule, we can conclude that approximately 68% of American women have shoe sizes between 6.54 and 9.58.

Using the empirical rule, we can say that approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.

To find the percentage of American women who have shoe sizes between 6.54 and 9.58, we first need to standardize these values by subtracting the mean and dividing by the standard deviation:

z1 = (6.54 - 8.06) / 1.52 = -1.00

z2 = (9.58 - 8.06) / 1.52 = 1.00

Now, we can use the empirical rule to find the percentage of women with shoe sizes between z = -1.00 and z = 1.00. Since this interval corresponds to one standard deviation from the mean, we know that approximately 68% of the data falls within this range.

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Total hours handled by x judges in the year ≤ 12120x where x is the number of judges needed to handle the caseload for the year.

To formulate an LP for this problem, we can define the decision variables and the objective function and constraints.

Decision Variables:

Let x be the number of judges needed by the city to handle the caseload for the year.

Objective Function:

We want to minimize the number of judges needed, so our objective function is:

minimize x

Constraints:

Each judge can handle up to 120 hours of caseload per month, so the total number of hours that can be handled by x judges in a month is 120x. The total number of hours needed to handle the caseload for the year is given in the table. Therefore, we have the following constraints:

For January: 120x ≥ 100

For February: 120x ≥ 120

For March: 120x ≥ 90

For April: 120x ≥ 80

For May: 120x ≥ 110

For June: 120x ≥ 130

For July: 120x ≥ 140

For August: 120x ≥ 110

For September: 120x ≥ 100

For October: 120x ≥ 90

For November: 120x ≥ 80

For December: 120x ≥ 120

These constraints ensure that the total number of hours handled by the judges in each month is at least as much as the number of hours needed to handle the caseload in that month.

We also need to ensure that all cases are handled by the end of December. This can be expressed as follows:

Total hours handled by x judges in the year ≤ 12120x

This constraint ensures that the total number of hours handled by the judges in the year is less than or equal to the maximum possible number of hours that can be handled by x judges in a year.

Therefore, the LP can be formulated as follows:

minimize x

subject to:

120x ≥ 100

120x ≥ 120

120x ≥ 90

120x ≥ 80

120x ≥ 110

120x ≥ 130

120x ≥ 140

120x ≥ 110

120x ≥ 100

120x ≥ 90

120x ≥ 80

120x ≥ 120

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The volume of a right circular cylinder with a diameter of 10 meters and a height of 16 meters is 400π m³.

Volume is the amount of space occupied by an object in three dimensions. It is a measure of how much space is contained within a solid shape, such as a cube, sphere, or cylinder. The unit of volume can vary based on the system of measurement being used, but common units include cubic meters, cubic centimeters, liters, and gallons. Volume is calculated by multiplying the area of the base of the object by its height or depth, depending on the shape of the object.

The volume of a right circular cylinder is given by the formula V = πr²h, where V is the volume of the cylinder, r is the radius of the circular base, and h is the height of the cylinder.

In words, the volume of a right circular cylinder is equal to the area of the circular base (πr²) multiplied by the height (h). This formula holds true regardless of the dimensions of the cylinder, as long as it has a circular base and straight sides that are perpendicular to the base.

The volume of a right circular cylinder is given by the formula V = πr²h, where r is the radius of the circular base and h is the height of the cylinder.

Since the diameter of the circular base is 10 meters, the radius is half of that, or 5 meters. The height of the cylinder is 16 meters.

Substituting these values into the formula, we get:

V = π(5)²(16) = 400π

Therefore, the volume of the cylinder is 400π cubic meters.

So the answer is 400π m³.

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The equation in standard form is x - 9y = -45.

The equation of a line in standard form is Ax + By = C, where x and y are variables and A, B and C are integers.

Given equation is y = 1/9 x + 5.

Here 1/9 is not an integer.

So multiplying the equation throughout by 9, we get,

9y = x + (5 × 9)

9y = x + 45

Subtracting both sides by x,

9y - x = 45

-x + 9y = 45

Multiplying throughout by -1,

x - 9y = -45, which is in standard form.

Hence the equation is x - 9y = -45 in standard form.

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The standard form of the equation of the quadratic function is y = -1/8x^2 + 1/4x - 41/8

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

(h, k) = (1, -5)

(x, y) = (5, -7)

A quadratic function is represented as

y = a(x - h)^2 + k

So, we have

y = a(x - 1)^2 - 5

Given that

(x, y) = (5, -7)

We have

-7 = a(5 - 1)^2 - 5

This gives

16a = -2

So, we have

a = -1/8

Recall that

y = a(x - 1)^2 - 5

So, we have

y = -1/8(x - 1)^2 - 5

Expand

y = -1/8(x^2 - 2x + 1) - 5

This gives

y = -1/8x^2 + 1/4x - 41/8

So, the required equation is y = -1/8x^2 + 1/4x - 41/8

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Answer:

26 meters,

hope this helps!

Answer: 38.8°

Step-by-step explanation:

See the attached diagram:

From it we can tell that the tunnel forms a right triangle, with the length of a leg and the hypotenuse given. We can use the sine function to easily answer this, (letting θ be the angle we need to find)

sinθ = 52/83

.: θ = arcsin(52/83)

(Note: arcsin is the inverse of the sine function)

.: θ ≅ 38.8°

The set of parametric equations are

[tex]x=\sqrt{t}[/tex]

[tex]y=\sqrt{3+t[/tex]

A parametric equation uses a parameter, which is an independent variable (often represented by the notation t), and dependent variables, which are expressed as continuous functions of the parameter and are independent of other variables.

Using one or more independent variables known as parameters, a parametric equation defines a set of quantities as functions.

Given the equation:

[tex]x= \sqrt[]{y^{2}-3 }[/tex]

Let's find a set of parametric equation for the given equation given the parameter:

[tex]x = \sqrt{t}[/tex]

From the parameter:

[tex]x = \sqrt{t}[/tex]

Now, substitute [tex]\sqrt{t}[/tex] for x in the given equation:

[tex]x=\sqrt{y^2-3} \\\\\sqrt{t} = \sqrt{y^2 - 3} \\\\t=y^2-3\\\\y^2=3+t\\\\y=\sqrt{3+t}[/tex]

Therefore, the set of parametric equations is:

• [tex]x=\sqrt{t}[/tex]

• [tex]y=\sqrt{3+t}[/tex]

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Explanation:

The normal curve, aka bell curve, has mirror symmetry with the mean at the very center. Half of the data is below the mean, while the other half is above the mean.

About 68% of the data will lie below the mean in a normally distributed data set. This is known as the Empirical Rule, which states that if a data set is approximately normally distributed, then approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

The irrational number for this problem are given as follows:

pi/2.

Rational numbers are numbers that can be represented by fractions, such as numbers that have no decimal parts, or numbers in which the decimal parts are terminating or repeating.

Irrational numbers are numbers that cannot be represented by fractions, being non-terminating and non-repeating decimals, such as non-exact square roots.

For this problem, we have that the number pi/2 is irrational, as pi is a non-terminating decimal, and the division of an irrational by a rational results in an irrational.

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The equation to represent the situation where the oldest child is 7 years older than the youngest child, using x for the age of the youngest child and y for the age of the oldest child, is y = x + 7.

If we let x represent the age of the youngest child in a family, then we can say that the oldest child is 7 years older than the youngest child. We can represent the age of the oldest child using the expression "x + 7", because if we add 7 to the age of the youngest child, we get the age of the oldest child.

Therefore, we can write an equation to represent this situation by equating the age of the oldest child to the age of the youngest child plus 7. This can be written as:

x + 7 = Oldest child's age

where x is the age of the youngest child, and "Oldest child's age" represents the age of the oldest child in the family. This equation tells us that if we know the age of the youngest child, we can find the age of the oldest child by adding 7 to it.

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Answer:

118

Step-by-step explanation:

Let the original number be x.

After subtracting 3, the result is x - 3.

When this result is multiplied by 5, we get (x - 3) * 5 = 575.

Expanding and solving for x, we have:

5x - 15 = 575

5x = 590

x = 118

So the original number was 118.

The upper quartile score on the final exam is 83.

Upper quartile, represented as Q₃, also known as third quartile is the value in a data set such that 75% of the points in the data set are below this value, after arranging in an ascending order.

Given data set is the scores for the final exam in an 8th grade math class.

58, 72, 74, 92, 84, 40, 74, 81, 76, 83.

First arrange the data in an ascending order.

40, 58, 72, 74, 74, 76, 81, 83, 84, 92.

Median is the middle element.

Median = (74 + 76) / 2 = 75

Upper quartile is the median of the second half.

Consider the second half.

76, 81, 83, 84, 92.

Median = 83

Third quartile = 83

Hence 83 is the upper quartile score on the final exam.

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Answer: Let's call the side length of the square pool "s". We know that the area of a square is equal to the length of one side squared, so we can set up an equation:

s^2 = 144

We can solve for s by taking the square root of both sides:

s = sqrt(144) = 12

So the side length of the pool is 12 feet.

The outside perimeter of the deck is equal to the perimeter of the square pool plus the width of the deck on each side (7 feet). So the outside perimeter is:

12 + 7 + 7 + 12 + 7 + 7 = 52 feet

So the outside perimeter of the deck is 52 feet.

Step-by-step explanation:

Answer: 52 feet

Step-by-step explanation:

First, we know this is a square. So, if the area is 144, square root of 144 is 12 (12 * 12 = 144)

Adding a width of 7 feet to each side means that we can add 12+12+7+7+7+7.

52 feet!

(っ◔◡◔)っ ♥

Hope this helps.

MM4343

Answer:

Step-by-step explanation:

Let's call the length of each shorter piece of the first pipe x. Then the total length of the first pipe would be 5x + 27 cm.

Similarly, the length of each shorter piece of the second pipe would be x and the total length of the second pipe would be 11x + 15 cm.

Since both pipes were cut into the same length pieces, we can set the two equations equal to each other:

5x + 27 = 11x + 15

Solving for x, we can find the length of each shorter piece:

6x = -12

x = -2

This means that each shorter piece of both pipes was -2 cm long, which is not physically possible. So, this solution is not valid. This means that the problem is underdetermined and that we cannot determine the length of each piece.

Answer:

5

Step-by-step explanation:

The rule of the sides of a triangle is that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This rule is also known as the triangle inequality theorem.

The formula to describe the variation is F = 1.6d².

Option C is the correct answer.

An equation is made up of one or two variables with a constant connected by an equal sign.

Example:

2x + 4 = 9 is an equation.

We have,

F is directly proportional to the square of d.

This can be written as,

F ∝ d²

F = kd² ____(1)

Now,

F = 40 when d = 5.

40 = k x 5²

k = 40/25

k = 8/5 _____(2)

Now,

From (1) and (2).

F = (8/5)d²

F = 1.6d²

Thus,

The formula to describe the variation is F = 1.6d².

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The ratio of the coordinate point on the line is  2 to 7.

The ratio of the coordinate point on the line is calculated as follows;

Let the coordinate points = (xp, yp)

Let the ratio of the line = a:b

xp = x₁  +  (a/(a+b) (x₂ - x₁)

yp = y₁  + (a/a+b) (y₂ - y₁)

-3 = -7 + ( a/a+b)(11 - - 7)

-3 = -7 + (a/a+b)(18)

-3(a + b)  = -7(a+b) + 18a

7(a + b ) - 3(a+b) = 18a

7a + 7b - 3a - 3b = 18a

4b = 14a

4/14 = a/b

2/7 = a/b

2 : 7 = a : b

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Answer: z is in the triangle  all triangles -180 degrees so 108+40 = 148

180-148=32
Answer 32 degrees

Step-by-step explanation:

Answer: 20 for acellus