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Question 4(Multiple Choice Worth 1 points)
(08.07 MC)

The quadratic function f(x) has roots of 3 and 7, and it passes through the point (1, 12). What is the vertex form of the equation of f(x)?

f(x) = −(x + 5)2 − 4
f(x) = −(x − 5)2 − 4
f(x) = (x + 5)2 − 4
f(x) = (x − 5)2 − 4

The equation of the linear function represented by the table in slope-intercept form is:

y = 3x + 2

We can find the equation of the linear function represented by the table using the slope-intercept form:

y = mx + b

where "m" is the slope of the line and "b" is the y-intercept.

To find the slope, we can use any two points from the table. Let's use the first and last points:

slope (m) = (y2 - y1) / (x2 - x1)

slope (m) = (14 - 5) / (4 - 1)

slope (m) = 3

Now we can find the y-intercept "b" by substituting one of the points and the slope into the slope-intercept form:

y = mx + b

5 = 3(1) + b

5 = 3 + b

b = 2

So the equation of the linear function represented by the table in slope-intercept form is:

y = 3x + 2

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Answer: c. 9

Step-by-step explanation:

In triangle ABC

a) The value of x = 29⁰

b) The angle c equal to 93⁰

A triangle is a closed plane figure that is formed by connecting three line segments, also known as sides, at their endpoints. The three endpoints, or vertices, where the sides of the triangle meet are not collinear. Triangles are important in mathematics and geometry because they are the simplest polygon that can exist in two-dimensional space.

In a triangle, the sum of all interior angles is always 180 degrees. Therefore, we can use this fact to find the value of x and angle c.

We know that:

angle a = 35⁰

angle b = 52⁰

angle c = 3(x+2)⁰

Using the fact that the sum of all interior angles in a triangle is 180 degrees, we can write:

angle a + angle b + angle c = 180

Substituting the values we know, we get:

35 + 52 + 3(x+2) = 180

Simplifying the equation, we get:

87 + 3x + 6 = 180

3x + 93 = 180

3x = 87

x = 29

Therefore, x = 29⁰

To find angle c, we can substitute the value of x into the equation we were given for angle c:

angle c = 3(x+2)

angle c = 3(29+2)

angle c = 3(31)

angle c = 93

Therefore, angle c is equal to 93⁰.

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Answer:

[tex]25.5h = 153[/tex]

[tex]h = 6[/tex]

The height is 6 feet, so A is correct.

The distance PQ cannot be expressed as k√5, where k is a rational number given that PQ = √18

To find dy/dx, we can use implicit differentiation:

4xy = 2(x² + 4y²) - 9x

Differentiating both sides with respect to x, we get:

4y + 4xy' = 4x + 8yy' - 9

Simplifying and solving for y', we get:

y' = (4x - 4y - 9)/(4x - 16y)

Hence, dy/dx = (4x - 4y - 9)/(4x - 16y).

At point P, the tangent to the curve is parallel to the y-axis.

This means that dx/dy = 0 at P.

We can find the coordinates of P by solving the following equations simultaneously

4xy = 2(x² + 4y²) - 9x

dx/dy = 0

From 4xy = 2(x² + 4y²) - 9x, we have:

4xy + 9x = 2(x² + 4y²)

Dividing both sides by x², we get:

4y/x + 9/x = 2 + 4y²/x²

Since dx/dy = 0, we have:

dy/dx = 0

Using the formula for dy/dx that we derived in part (a), we get:

(4x - 4y - 9)/(4x - 16y) = 0

Solving for y, we get:

y = (4x - 9)/4

Substituting this into 4xy = 2(x² + 4y²) - 9x, we get:

4x * (4x - 9)/4 = 2(x² + 4[(4x - 9)/4]²) - 9x

Simplifying and solving for x, we get:

x = 9/2 and 3/2

Substituting these values of x into y = (4x - 9)/4, we get the coordinates of P:

y = 9/4 and y = -3/4

This means that

P(9/2, 9/4) and P(3/2, -3/4)

At point Q, the tangent to the curve is parallel to the x-axis.

This means that dy/dx = 0 at Q.

Using the formula for dy/dx that we derived in part (a), we get:

(4x - 4y - 9)/(4x - 16y) = 0

Solving for x, we get:

x = (4y + 9)/4

Substituting this into 4xy = 2(x² + 4y²) - 9x, we get:

4 * [(4y + 9)/4] * y = 2([(4y + 9)/4]² + 4y²) - 9 * (4y + 9)/4

Simplifying and solving for y, we get:

y = -3/4, 9/4

Substituting these values of y into x = (4y + 9)/4, we get the coordinates of Q:

x = 3/2 and x = 9/2

This means that

Q(3/2, -3/4) and Q(9/2, 9/4)

Using the distance formula, we can find the distance PQ:

PQ = √[(9/2 - 3/2)² + (9/4 + 3/4)²] or √[(3/2 - 9/2)² + (-3/4 - 9/4)²]

Simplifying, we

PQ = √18 or √18

So, we have

PQ = √18

This gives

k√5 = √18

So, we have

k = √18/√5

Evaluate

k = √3.6

√3.6 is not a rational number

Hence, the distance PQ cannot be expressed as k√5

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Answer:

height= 3cm

Step-by-step explanation:

to find the base of each triangle just half the bottom to get 4cm

a^2+b^2=c^2

a^2+4^2=5^2

a^2+16=25

a^2=9

a=√9

a=3

According to the question the volume formula to find the height of the pyramid is 9 units.

Formula is a mathematical rule or equation expressed symbolically, that describes the relationship between different variables. It is a set of symbols or terms that expresses a relationship between different entities, such as numbers, objects, ideas, or concepts. Formulas are used to calculate, analyze, and predict data, and are essential in many scientific and mathematical fields.

The volume of a pyramid is given by the formula V = (1/3) × b × h, where b is the area of the base and h is the height of the pyramid.
Since the base of the pyramid is a right triangle, we can use the formula to find the area of a right triangle, A = (1/2) × b × h, to find the base of the pyramid.
A = (1/2) × 8 × 9 = 36
Now we can use the volume formula to find the height of the pyramid:
V = (1/3) × 36 × h

108 = (1/3) × 36 × h

108 = 12 × h

h = 9 units

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Answer:

28°

Step-by-step explanation:

If the angles are supplementary, they add up to 180°.

(3x) + (2x + 40) = 180

5x + 40 = 180

     - 40   - 40

5x = 140

x = 28°

The final value of the annuity due is N165,603.

A series of equal payments paid at regular intervals over a certain time period is known as an annuity. Because they enable people to save a specific sum of money over time and receive consistent payments in the future, annuities are frequently utilised in finance, particularly for retirement planning. The frequency of payments for an annuity might be monthly, quarterly, semi-annually, or annually, and they can be set or variable. Depending on when the payments start, annuities can also be categorised as immediate or deferred.

The effective monthly rate is given as:

[tex]i = (1 + 0.09/12)^{12} - 1 = 0.007434[/tex]

Now, the future value is given by:

[tex]FV = 10000 * ((1 + i)^{120} - 1) / i = 1,473,820.76[/tex]

For annuity due add one payment more to get the value:

FV = FV * (1 + i) = 1,490,304.84

Hence, the final value of the annuity due is N165,603.

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Answer: To calculate how many times Francisco will earn interest after keeping the money in the account for 5 years, we need to determine the number of quarters in 5 years.

Since there are 4 quarters in a year, the total number of quarters in 5 years is:

4 quarters/year x 5 years = 20 quarters

Therefore, Francisco will earn interest 20 times over the 5-year period.

To calculate the value of his account in 5 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the final amount

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time the money is invested (in years)

In this case, P = $5,000, r = 0.03, n = 4 (since the interest is compounded quarterly), and t = 5. Substituting these values into the formula, we get:

A = $5,000(1 + 0.03/4)^(4x5)

A = $5,000(1.0075)^20

A = $5,866.02

Therefore, the value of Francisco's account in 5 years, if he does not make any additional deposits or withdrawals, will be $5,866.02.

Step-by-step explanation:

The total surface area that needs painting is approximately 487 square inches.

What is the cone total surface area?

The total surface area (TSA) of a cone is given by the formula:

TSA = πr(r + l)

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

[tex]Surface Area = \pi r(r + \sqrt{(r^2 + h^2)} )[/tex]

where r is the radius of the base of the cone and h is the slant height of the cone.

Given:

Diameter of the cone = 12 inches

Radius (r) = Diameter / 2 = 12 / 2 = 6 inches

Slant height (h) = 19 inches

Plugging in the values into the formula, we get:

[tex]Surface Area = \pi * 6 * (6 + \sqrt{(6^2 + 19^2)} )[/tex]

Calculating the value inside the square root:

[tex]\sqrt{(6^2 + 19^2)} = \sqrt{(36 + 361) } = \sqrt{397} = 19.92[/tex] (rounded to two decimal places)

Substituting the value back into the formula:

[tex]Surface Area = \pi * 6 * (6 + 19.92) = \pi * 6 * 25.92 = 487.34[/tex] square inches (rounded to two decimal places)

Hence, the total surface area that needs painting is approximately 487 square inches.

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Answer: a. To find the number of barrels that should be refined to maximize profit, we need to find the critical point of the function P(x), which occurs where the derivative of P(x) equals zero.

P(x) = 8x - 0.02x²

P'(x) = 8 - 0.04x

Setting P'(x) = 0, we get:

8 - 0.04x = 0

Solving for x, we get:

x = 200

Therefore, 200 barrels should be refined to maximize the profit.

b. To find the maximum profit, we substitute x = 200 into the profit function P(x):

P(200) = 8(200) - 0.02(200)²

P(200) = 1600 - 800

P(200) = 800

Therefore, the maximum profit is $800.

Step-by-step explanation:

still 7 inches since translation and rotation do not affect area, angles and the length of the sides. Therefore, if you look at the angles, CA will be the same as DF

The function has a vertical asymptote at x = 1, x-intercepts are x = 1 and x = 5, hole at x = 5 and horizontal asymptote is y = 0.

A rational function is a function that can be expressed as the ratio of two polynomial functions. In other words, it is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions and q(x) is not equal to zero for any value of x.

To plot the rational function f(x) = (x² - 6x + 5)/(-x + 1)

Vertical asymptote: The denominator of the function (-x + 1) is equal to zero when x = 1.

Therefore, there is a vertical asymptote at x = 1.

x-intercept: To find the x-intercept, we set the numerator equal to zero and solve for x:

x² - 6x + 5 = 0

This quadratic equation can be factored as:

(x - 5)(x - 1) = 0

Therefore, the x-intercepts are x = 1 and x = 5.

y-intercept: To find the y-intercept, we set x equal to zero:

f(0) = (0² - 6(0) + 5)/(-0 + 1) = 5

Therefore, the y-intercept is (0, 5).

Hole: The function has a hole at x = 5 because both the numerator and the denominator become zero at x = 5.

Horizontal asymptote: To find the horizontal asymptote, we need to compare the degrees of the numerator and the denominator. The degree of the numerator is 2 and the degree of the denominator is 1, so the horizontal asymptote is y = 0.

Now, we can plot the function by choosing some values of x and calculating the corresponding values of y:

x y = f(x)

-1 4

0 5

0.5 3.25

1 undefined

2 -1

Image is attached below.

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Thus, the unknown side length of rectangle for the given area and one side length is found to be 5.5 m.

Every day, rectangles are all around us. All of these objects—doors in a home, windows in a structure, and books in a library—are rectangles.

Given that:

Area of rectangle A = length * width

width = area / length

width = 11 / 2

width =  5.5

Thus, the missing side of rectangle for the given area and one side length is found to be 5.5 m.

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Complete question:

Find the unknown side length of the rectangle if its area is 11 m² and one of its side is 2 m.

Answer: 3

Step-by-step explanation:

Using pemdas we know that the parrenthesis comes first. I will do 10-1=9. Now, I need to divide it by three. 9/3=3. 3 is the answer

The probability of exactly 3 successes in 15 trials with a probability of success on a single trial of 0.77 is approximately 0.1648.

To find the probability of exactly 3 successes in a binomial distribution with n=15 trials and a probability of success on a single trial q=0.77, we can use the formula for the probability mass function of the binomial distribution:

P(X=k) = (n choose k) * q^k * (1-q)^(n-k)

Substituting the given values into the formula, we get:

P(X=3) = (15 choose 3) * 0.77^3 * (1-0.77)^(15-3)

P(X=3) = (15!/((15-3)!3!)) * 0.77^3 * 0.23^12

P(X=3) = 0.1648.

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The three consecutive integers are 18, 19, and 20.

Consecutive integers are integers that follow each other in order, without any gaps, and differ by 1. For example, 3, 4, and 5 are consecutive integers because they follow each other in order and differ by 1.

Let's assume that the three consecutive integers are x, x+1, and x+2. Then, according to the problem statement:

4x + (x+2) = 92

Simplifying this equation, we get:

5x + 2 = 92

Subtracting 2 from both sides, we get:

5x = 90

Dividing both sides by 5, we get:

x = 18

Therefore, the three consecutive integers are 18, 19, and 20. We can check that these integers indeed satisfy the condition given in the problem:

4 times the first integer is 4*18 = 72, and when we add the third integer 20 to it, we get 72 + 20 = 92, as required.

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The radius of the cylinder is approximately 6.75 inches.

We can use the formula for the volume of a cylinder to solve this problem:

[tex]V = \pi r^2h[/tex]

We know that the volume V is 2,035.75 [tex]in^3[/tex], and the height h is 3 times the radius r. So we can write:

[tex]V = \pi r^2(3r)[/tex]

Simplifying this expression, we get:

V = 3π[tex]r^3[/tex]

To solve for r, we can divide both sides of the equation by 3π[tex]r^2[/tex]:

V/(3π[tex]r^2[/tex]) = r

Substituting the given value for V, we have:

2,035.75/(3π[tex]r^2[/tex]) = r

Multiplying both sides by 3πr^2, we get:

2,035.75 = 3π[tex]r^3[/tex]

Dividing both sides by 3π, we have:

[tex]r^3[/tex] = 2,035.75 / (3π)

Taking the cube root of both sides, we get:

r = [tex](2,035.75 / (3\pi ))^{1/3[/tex]

Using a calculator, we find that:

r ≈ 6.75 inches

Therefore, The cylinder has a radius of about 6.75 inches.

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Thus, the height of flagpole from the ground is found as 23.34 feet.

The measurement of the angle between a person's eyes' line of sight to anything above and the horizontal line is known as the angle of elevation.

The movement of the observer's eyes determines the elevation angle. The angle of elevation is the angle formed by the line of sight and the horizontal line when a viewer is looking up at an object.

Given data:

Using the trigonometric ratios in the right triangle ABE.

tan 35 = h / 30

h = 30* tan(35)

h = 18.34

Height of flagpole = 18.34 + 5 = 23.34 feet

Thus, the height of the flagpole from the ground is found as 23.34 feet.

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Equation of circle in standard form- (x + 1)² + (y + 2)² = 1².  radius of the circle is found as 1. centre -(h,k) =  (-1, -2)

We can solve quadratic equations that lack a factor by completing square.

In order to make the left side of an equation a perfect square trinomial, the equation's form must be adjusted.

Given equation is-

x² + y² + 2x + 4y + 4 = 0

Add and subtract 1 in the equation:

x² + y² + 2x + 4y + 4 + 1 - 1 = 0

rearranging:

(x² + 2x + 1) + (y² + 4y + 4) - 1 = 0

This can be written as:

(x² + 2*1x + 1²) + (y² + 2*2y + 2²) = 1

(x + 1)² + (y + 2)² = 1²  ...eq 1

Thus, the standard form of the equation of circle is:

(x - h)² + (y - k)² = r²  ..eq 2

On comparing eq 1 and eq 2

In which radius of the circle is found as 1.

centre -(h,k) =  (-1, -2)

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The finance charge for Antoni Gaudi's bedroom set purchase is $731.08.

An installment refers to a sum of money that is paid back in parts over a period of time, typically with interest.

A loan is a financial agreement between a lender and a borrower, in which the lender provides the borrower with a specific amount of money or assets, which the borrower agrees to pay back with interest over a predetermined period of time. Loans can be used for a variety of purposes, such as purchasing a home, starting a business, or paying for education.

Based on the given information, the bedroom set sells for $2,650, and the financing requires a 15% down payment, which is $397.50 ($2,650 x 0.15).

Therefore, the amount financed is $2,252.50 ($2,650 - $397.50).

To calculate the finance charge, we need to know the total amount of payments Antoni Gaudi will make over the 42-month period. Each monthly payment can be calculated using the following formula:

Monthly Payment = (Amount Financed x (APR/12)) / (1 - (1 + APR/12)^-n)

Where:

APR is the annual percentage rate (12% in this case)

n is the total number of payments (42 in this case)

Plugging in the values, we get:

Monthly Payment = ($2,252.50 x (0.12/12)) / (1 - (1 + 0.12/12)^-42)

= $70.99 (rounded to the nearest cent)

Therefore, the total amount of payments Antoni Gaudi will make over the 42-month period is $2,983.58 ($70.99 x 42).

The finance charge can be calculated as follows:

Finance Charge = Total Payments - Amount Financed

= $2,983.58 - $2,252.50

= $731.08

Therefore, the finance charge for Antoni Gaudi's bedroom set purchase is $731.08.

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The unit twenty-six point seven ounces in terms of milligrams is 756932.3  

To convert ounces (oz) to milligrams (mg), we can use the following conversion factor:

1 oz = 28,349.5 mg

Therefore, to convert 26.7 oz to mg, we can multiply 26.7 by the conversion factor as follows:

26.7 oz × 28,349.5 mg/oz = 756932.3  mg

So, there are approximately 756932.3 milligrams in 26.7 ounces of mass.

Comparing the calculated answer with the options given, we see that the answer is closest to 756,498.15 mg, which is not one of the options.

Therefore, we cannot choose any of the options given.

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Answer:

mean: 20mins

Step-by-step explanation:

add them all together to get 240

then divide by how many there are

240÷12=20

so the mean is 20mins

The product of the binomial (d-1)(d+2)(d-3) is d³-2d²-5d+6.

A binomial is an expression with two terms. For example 3x+4 is an example of binomial and also 3x²+3x is also an example of binomial.

To find the product of (d-1)(d+2)(d-3) , we multiply the three binomial together to give us a polynomial expression.

(d-1)(d+2) = d(d+2) -1( d+2)

= d²+2d-d-2

= d²+d-2

now solving (d²+d-2)(d-3)

= d( d²+d-2) -3(d²+d-2)

= d³+d²-2d-3d²-3d+6

collect like terms

= d³+d²-3d²-2d-3d+6

= d³-2d²-5d+6

therefore the product of (d-1)(d+2)(d-3) is d³-2d²-5d+6

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Answer:

It is irrational, because the product of an irrational number and a rational number is irrational.

Step-by-step explanation:

sqrt 27 is  irrational =5.19615.......

3 is rational

  subtracting 3 from the irrational number just creates a smalller irrational number    ....then multiplying by 9 just creates a larger irrational number