Write and solve a polynomial equation for the situation described.
A rectangular two-story horse barn is
being designed for a farm. The upper floor will be used
for storing hay, and the lower floor will have horse stalls
that extend 5 feet from both of the longer walls. The barn’s
length is twice the barn’s width, and the lower floor’s
ceiling height is 6 feet less than the barn’s width. What
should the dimensions of the lower floor be if the space
not used for stalls is to have a volume of 1920 cubic feet?

If you travel at a constant speed, you follow this equation:

[tex] d = st [/tex]

where d is the distance, s is the speed and t is the time.

In this example, you first travel 60 miles per hour for 45 minutes. Since 45 minutes are 3/4 of a hour, you will cover a distance of

[tex] d_1 = 60\cdot\dfrac{3}{4} = 45 [/tex]

In the second part, since 30 minutes is half a hour, you will cover a distance of

[tex] d_2 = 50\cdot\dfrac{1}{2}=25 [/tex]

So, in total, you travel

[tex] d=d_1+d_2=45+25 = 70 [/tex] miles

Answer:

C-70 miles

Hope this helped :)

Answer:

$9800

Step-by-step explanation:

commission = 5% × car price

20 × commission = 100% × car price . . . . . . multiply by 20

20 × $490 = car price = $9800 . . . . . . . . . . fill in given value of commission

The price of a car on which a salesperson earns a 5% commission of $490 is calculated by dividing the commission amount by the commission rate as a decimal (0.05), resulting in a car price of $9,800.

The question is asking to calculate the price of a car based on the commission earned by the salesperson. The salesperson receives a 5% commission on each sale and has earned $490 from the sale of one car.

To find the price of the car, we need to set up a simple equation. If 5% of the price results in a $490 commission, we need to divide the commission by the percentage rate expressed as a decimal.

The calculation will look as follows:

Convert the commission percentage to a decimal: 5% = 0.05.

Divide the commission earned by the decimal rate: $490 ÷ 0.05.

The result is the price of the car.

So, $490 ÷ 0.05 = $9,800.

Therefore, the price of the car is $9,800.

Answer:

x = 3

Step-by-step explanation:

f'(x) = 0 for x = 3 and x = 4 . . . . by the zero product rule.

The coefficient of x² in f'(x) is negative, so the parabola opens downward.

f''(x) is positive for x < 3.5, so the coordinate x = 3 represents a relative minimum.

See the attachment

In this context, "x" stands for "a number". Then 1/2x is "one-half of a number" and 3/4x is "three-fourths of a number."

3/4 - ( ) means "three-fourths minus ..."

( ) + 1/2 means "one-half more than ..." or "the sum of ... and one-half."

Putting these ideas together lets you choose the phrases as indicated below.

[tex]0.5x+\dfrac{7}{8}=5\qquad\text{multiply both sides by 8}\\\\4x+7=40\qquad\text{subtract 7 from both sides}\\\\4x=33\qquad\text{divide both sides by 4}\\\\\boxed{x=\dfrac{33}{4}\to x=8\dfrac{1}{4}}[/tex]

14.4

This is a special case of the situation where two secant lines intersect a circle. The product of the near distance to the circle and the far distance to the circle from their point of intersection is a constant. A tangent is a degenerate case, where the near and far intersection points with the circle are the same point.

So, ...

... x · x = 9 · (9+14)

... x = 3√23 ≈ 14.4 . . . . . taking the square root

_____

Additional note on the geometry

This rule for the product of segments of secant lines also applies when the secants intersect inside the circle.

Answer:

14.39

Step-by-step explanation:

Theorem 12-15 III

(y+z) y=t squared

(9+14)9=x squared

23•9=x squared

207 =x squared

Square root of 207 is 14.39

Answer:

Table B shows a proportional relationship.

Step-by-step explanation:

In a proportional relationship two quantities vary directly with each other. It means

[tex]y\propto x[/tex]

[tex]y=kx[/tex]

Where, k is the constant of variation.

The ordered pairs of table A are (-2,2), (-1,3), (0,0) and (1,5).

From these ordered pairs we can conclude that the value of y-coordinate is not changing according to the x-coordinate because the values of x increased by 1 for each ordered pair but the value of y is not increasing in the same proportion..

The ordered pairs of table B are (-1,-3), (0,0), (1,3) and (2,6). The value of y increasing at a constant rate 3 and the value of y-coordinate is 3 times of x-coordinate.

Choose any two ordered pairs of table B. Let the two points are (0,0) and (1,3), then the constant of variation is

[tex]k=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{1-0}=3[/tex]

The proportional relationship is defined as

[tex]y=3x[/tex]

Therefore, 3 is the constant of variation and rate of change.

So, Table B shows a proportional relationship.

Answer:

Table B Shows proportional relationship.

Step-by-step explanation:

Correct answers are the green and yellow dots. Hope this helps. The red dot I got wrong.

Answer:

C. The length of one lap.

Step-by-step explanation:

In the first four words, the problem statement tells you that L is the number of laps run. It is not the length of one lap.

_____

Comment on the problem presentation

Appropriate punctuation would be very helpful. Apparently, it is a 1/4-mile track, not a 14-mile track. Apparently, the distance is 1 1/2 miles, not 112 miles. Copying and pasting problem text often leaves out the special symbols used on some web sites. Some editing is usually needed.

Answer:

Step-by-step explanation:

A graphing calculator can show you the intercepts.

___

Or you can figure them out.

The y-intercept is where x=0, so is ...

  y = log(12·0 +7) -3 = log(7) -3 ≈ -2.1549 ≈ -2.15

___

The x-intercept is where y = 0, so is ...

  0 = log(12x +7) -3

  3 = log(12x +7) . . . . . . add 3

  10^3 = 12x +7 . . . . . . . take the antilog

  993 = 12x . . . . . . . . . . subtract 7

  993/12 = x = 82.75 . . . divide by the coefficient of x

Answer:

68

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you ...

... Tan = Opposite/Adjacent

For this geometry, this means ...

... tan(x°) = 5/2

Taking the inverse tangent, we can find x.

... x° = arctan(5/2) ≈ 68.199°

... x ≈ 68

The polynomial function f(x) having roots [tex]-2 + \sqrt(8)[/tex] and 9 would have factors  [tex](x + 2 - 2\times \sqrt(2))[/tex] and (x - 9). These roots can be determined by setting the known values equal to x and subtracting from x.

In mathematics, specifically in polynomial functions, the roots of a function are values that make the function equal to zero. These roots correspond to the factors of the polynomial function. If a function f(x) has roots -2 + square root 8 and 9, then the factors of f(x) will be those values set equal to x and then subtracted from x.

Therefore, the corresponding factors would be  [tex]x - (-2 + \sqrt{8})[/tex] and (x - 9). When you simplify these, they become [tex](x + 2 - \sqrt(8))[/tex] and (x - 9). To further simplify, we know that  [tex]\sqrt(8) = 2\times \sqrt(2),[/tex] therefore we get  [tex](x + 2 - 2\times \sqrt(2))[/tex] and (x - 9).

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If a polynomial function has roots -2 + Square root 8 and 9, then x - (-2 + Square root 8) and x - 9 must be factors of the polynomial.

Polynomial functions are mathematical expressions comprising variables, coefficients, and non-negative integer exponents. They are used in various fields, including mathematics, physics, engineering, and computer science. They model relationships between variables, aiding in solving equations, analyzing data, and predicting outcomes in diverse applications.

If the polynomial function f(x) has roots -2 + Square root 8 and 9, then x - (-2 + Square root 8) and x - 9 must be factors of f(x). This is because if a number is a root of a polynomial, then the polynomial can be divided evenly by the corresponding linear factor.

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You want to determine the number of text messages students at your school send in a month. You randomly ask 10 students from each grade.

This is a random sampling method.

This sampling method is a basic sampling method where we select a group of subjects (a sample) for study from a larger group or  population. Each individual is chosen entirely by chance. Each member of the population has an equal chance of being included in the sample.

Answer:

answer is pretty much in the last sentance lol. random

Step-by-step explanation:

The numerator can be found by multiplying each of these fractions by 100. It can be helpful to reduce these fractions first.

Here, we calculate those numerators.

a) 15/60 · 100 = 1/4 · 100 = 100/4 = 25

b) 30/60 · 100 = 1/2 · 100 = 100/2 = 50

c) 25/75 · 100 = 1/3 · 100 = 100/3 = 33 1/3

Of course, the denominator is 100, so the fractions are ...

... 15/60 = 25/100

... 30/60 = 50/100

... 25/75 = (33 1/3)/100

_____

Comment on this solution

Effectively, we have multiplied each fraction by 1 = 100/100. That is ...

... fraction × (100/100) = (fraction × 100)/100

This changes the form, but not the value.

Answer:

the first selection (see below)

Step-by-step explanation:

The average rate of change (m) on the interval [x1, x2] is given by ...

... m = (g(x2) -g(x1))/(x2 -x1)

For g(x) = -x²-4x and (x1, x2) = (6, 8), the expression is the one attached.

Answer:

[tex]\frac{[-8^2 - 4(8)]- [-6^2 - 4(6)]}{8-6}[/tex]

Step-by-step explanation:

average rate of change of the function g(x) = -x^2 - 4x on the interval 6 ≤ x ≤ 8

To find average rate of change we use formula

Average =[tex]\frac{g(x_2)-g(x_1)}{x^2-x_1}[/tex]

use the given interval 6<=-x<=8

x2=8  and x1= 6

we replace the value in the given formula

g(x) = -x^2 - 4x

[tex]g(8) = -8^2 - 4(8)[/tex]

[tex]g(6) = -6^2 - 4(6)[/tex]

x2-x1 is 8-6

So equation becomes

[tex]\frac{[-8^2 - 4(8)]- [-6^2 - 4(6)]}{8-6}[/tex]

Answer:

64

Step-by-step explanation:

Since 64 is a multiple of itself and of all the other numbers, the answer is 64.

Answer:

$217.07

Step-by-step explanation:

Each year, the cost of the groceries was multiplied by 100% +2.7% = 1.027. Doing this multiplication for 5 years result in a multiplying facor of ...

... 1.027^5

so the cost of the groceries after 5 years is ...

... $190.00 × 1.027^5 ≈ $217.07

Answer:

4

Step-by-step explanation:

8 2/9 is close to 8

3 6/7 is close to 4

8 2/9 - 3 6/7  is close to 8-4 = 4

My estimate is 4

In 4 months, the total cost of Yoga Studio A would equal that of Yoga Studio B

The cost of Yoga Studio A at any given month can be represented as:

= Membership fee + (Monthly fee x Number of months)

Assuming number of months is x, the formula would be:

= 24 + (21.50 × x)

= 24 + 21.50x

Yoga Studio B would be:

= 41 + (17.25 × x)

= 41 + 17.25x

In order to find the month that these costs would be equal, equate both formulas:

24 + 21.50x = 41 + 17.25x

21.50x - 17.25x = 41 - 24

4.25x = 17

x = 17 / 4.25

x = 4 months

In conclusion, their costs would be the same in the 4th month.

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3x = -4y

To solve for X, divide both sides by 3:

x = -4y / 3

The answer is D.

Answer:

P

Step-by-step explanation:

The distance doesn't change during segment Q, so that is when the cyclist is stopped. The segment before that is labeled P.

The question asks the label of the segment before the one where the cyclist was stopped, so the appropriate choice is P.

Answer:

it is Q

Step-by-step explanation:

Because it is stopped and its a strait line making it look like it is taking a brake.

First differences start at 1 and increase by 2.

Whenever analyzing any sort of sequence, it is usually helpful to look at the differences between terms. Here, they are ...

... 1, 3, 5, 7

This set of numbers clearly increases by 2 from one to the next. That is, second differences are 2.

_____

When 2nd differences are constant, the pattern can be described by a 2nd-degree polynomial. In this case, that is

... n² -2n +2 . . . . for n = 1, 2, 3, ...

Answer:

Quadratic sequence with

nth term = n^2 - 2n + 2.

Step-by-step explanation:

1,2,5,10,17

The differences between terms  are

1,3,5,7

and the second differences are

2,2,2

This is a quadratic sequence  with first term n^2  ( where n = sequence number).

List the original sequence and the values of n^2:-

1  2  5  10  17

1  4  9  16  25

If we subtract the first list from the second we get

0  2  4  6  8   which is arithmetic with common difference  2 and first term 0

The nth term of this is 2(n - 1)

So the explicit formula for the nth term of the original sequence is

n^2 - 2(n - 1)

= n^2 - 2n + 2

Multiplying it out using the distributive property, you have ...

... x^4 -(x^4 -7x^2 +7x^2 -49)

... = x^4 -x^4 +7x^2 -7x^2 +49 . . . . distribute the minus sign

... = x^4(1 -1) +x^2(7 -7) +49 . . . . . . collect like terms

... = 0 +0 + 49 . . . . . . . . . . . . . . . . . .simplify

... = 49

50 people

Let x represent the number of people on the cruise. The amount they each must pay is ...

... ($58 -(x -42)) = $100 -x

The revenue from the group is the product of the number of people and the amount each pays:

... r(x) = x·(100 -x)

This describes a downward-opeing parabola with zeros at x=0 and x=100. The vertex (maximum) will be found halfway between those zeros, at x=50.

A group size of 50 maximizes revenue from the cruise.

Answer:

14 years 7 months

Step-by-step explanation:

If the average age was increased by 2 months by adding a 7th child, the total age was increased by 2·7 months = 14 months more than the previous average. That is, the added child's age is 14 months more than the previous average, so is ...

... (13 years + 5 months) + (1 year + 2 months) = 14 years + 7 months

_____

For the non-believers:

The previous total of all ages was ...

... 6 × (13 years 5 months) = 80 years 6 months

The new total of all ages is ...

... 7 × (13 years 7 months) = 95 years 1 month

The amount added to the previous total is ...

... (95 1/12 - 80 6/12) years = 14 7/12 years = 14 years 7 months

To find the age of the seventh child, calculate the total age of the six children and the total age of all seven children.

To solve this problem, we can use the concept of the average. Step 1: Calculate the total age of the six children by multiplying the average age (13 years and 5 months) by the number of children (6). Step 2: Calculate the total age of all seven children by multiplying the new average age (13 years and 7 months) by the total number of children (7). Step 3: Subtract the total age of the six children from the total age of all seven children to find the age of the seventh child.

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Answer:

y = ln(x -5)

Step-by-step explanation:

The log function has a vertical asymptote at x=0 and goes through the point (1, 0). This looks like that function translated 5 units to the right. When f(x) is translated right 5 units, x is replaced by x-5. The graph seems to be of the function ...

... y = ln(x -5)

_____

The natural log function will go through the points (2.718, 1) and (7.389, 2), as this curve appears to do.

Answer:

D. $5.37

Step-by-step explanation:

We need to figure out how much grapes cost per pound

Change the mixed number to an improper fraction

7 1/5 = (5*7+1) /5 = 36/5

$12.89 / 36/5

Copy dot flip

12.89 * 5/36 = 1.79 per lb

If we are purchasing 3 pounds, we will multiply by 3

3 lbs * $1.79 / lb = $5.37

Answer:

The probability is about 7.7%

Step-by-step explanation:

A probability is the ratio of the number of relevant outcomes and the number of all possible outcomes. To answer this question we do some counting first:

Consider two draws. We are interested in the second draw being an Ace, the first draw can either be an Ace or not, so, there are two cases of a relevant outcomes (all without replacement):

(number of relevant oucomes n ) = (number of cases of Ace-Ace draws) + (number of cases of NonAce-Ace draws):

[tex]n=4\cdot 3+{48\cdot4}= 204[/tex]

(number of possible outcomes) = (number of choices at first draw) * (number of choices at second draw) = 52 * 51 = 2652

The probability is then

P = 204/2652 = 0.0769, or about 7.7%

Answer:

y = 3x +8

Step-by-step explanation:

The slope of the given line is -1/3. The slope of a perpendicular line is the opposite of the reciprocal of that: -1/(-1/3) = 3.

Then, in point-slope form, the equation of the line is ...

... y - k = m(x - h) . . . . . for slope m through point (h, k)

... y - 5 = 3(x -(-1)) . . . . . for line of slope 3 through (-1, 5)

... y = 3x +8 . . . . . . . . . simplify

Step-by-step explanation:

1. See the attachment for the filled-in diagram. Adding the contents of the figure gives the sum at the bottom, matching selection C.

2. If we let "d" represent the length of the second volyage, then the total length of the two voyages is ...

... (d+43) + d = 1003

... 2d = 960 . . . . . . . subtract 43

... d = 480 . . . . . . . . divide by 2

The second voyage lasted 480 days.

3. 1.9% - 1.9/100 = 0.019. Adding this fraction to the original means the original is multiplied by 1 +0.019 = 1.019. Doing this multiplication each year for t years means the multiplier is (1.019)^t.

Since the starting value (in 1975) is 4 (billion), the population t years after that is ...

... P(t) = 4(1.019)^t

C. None of the above

It appears to me the average times for Ravens backs are lower (< 4.44 s) than those of Chiefs backs (< 4.48 s), so the Ravens are faster on average.

The spread of Ravens times is 4.30–4.57 s, whereas the spread of Chiefs times is 4.42–4.56. The Ravens clearly show a greater variation.

The offered choices suggest the Chiefs are faster on average (not true) and that the Chiefs have a greater variation (not true). Since both of these choices are incorrect, the only remaining suitable choice is None of the above.