at a local department store shirts were sold for 15 each.
This price was 80% of their regular price. What was the original price?

The Rational Roots Theorem states that the possible rational roots of a polynomial equation are the factors of the constant term divided by the factors of the leading coefficient. Therefore, the possible rational roots of P(x) are ±1, ±2, ±4, and ±4.

To find the exact roots of P(x), we can use synthetic division. Synthetic division is an efficient way of dividing a polynomial by a number. Using this method, we can determine that there are three roots for P(x): 1, -2, and -3. To verify this, we can evaluate P(x) for each of the three roots and confirm that the result is zero.

We can also use the quadratic formula to find the remaining two roots. Since P(x) is a 5th degree polynomial, the two remaining roots are complex and can be found using the quadratic formula. The complex roots of P(x) are -0.5 ± 0.5i.

In conclusion, the roots of P(x) are 1, -2, -3, -0.5 ± 0.5i. All of these results can be verified by evaluating P(x) and confirming that the result is zero.

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Answer:

We need to find out how many trips Billy needs to make to transport 80 cubic meters of mulch. Since we know the dimensions of the cargo bed, we can find its volume and then divide the total volume of mulch by the volume of the cargo bed to find the number of trips.

The volume of the cargo bed is:

V = l × w × h

where l is the length, w is the width, and h is the height. Using the dimensions given in the diagram, we get:

V = 4 m × 2.5 m × 1.5 m = 15 cubic meters

Now we can divide the total volume of mulch by the volume of the cargo bed to find the number of trips:

Number of trips = total volume of mulch / volume of cargo bed

Number of trips = 80 cubic meters / 15 cubic meters ≈ 5.33

Since Billy cannot make a fractional number of trips, he will need to make 6 trips to transport all the mulch to his house.

The hypotenuse is sqrt(11^2 + 7^2) = sqrt(170). Then, sin t = 11/sqrt(170), cos t = 7/sqrt(170), csc t = sqrt(170)/11, sec t = sqrt(170)/7, and cot t = 7/11.

Since we know that tan t = 11/7, we can use the Pythagorean identity (sin^2 t + cos^2 t = 1) to find the other trigonometric functions. First, we will find sin t and cos t:

tan t = 11/7 = opposite/adjacent = sin t/cos t

sin t = 11*cos t

cos t = 7*sin t

Substituting the second equation into the first equation:

sin t = 11*(7*sin t)

sin^2 t = 121*sin^2 t

121*sin^2 t - sin^2 t = 0

120*sin^2 t = 0

sin^2 t = 0/120

sin^2 t = 0

sin t = 0

Since sin t = 0, cos t = 1. Now we can find the other trigonometric functions:

csc t = 1/sin t = 1/0 = undefined

sec t = 1/cos t = 1/1 = 1

cot t = 1/tan t = 1/(11/7) = 7/11

So, the values of the trigonometric functions are:

sin t = 0

cos t = 1

csc t = undefined

sec t = 1

cot t = 7/11

Note: Another way to find the values of the trigonometric functions is to use the Pythagorean Theorem to find the hypotenuse of the right triangle with sides 11 and 7.

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Therefore, the best statement supported by the information in the box plots is statement B, "the data for Mr. Flores’s class are more symmetrical than the data for Mrs. Morales's class."

A box plot, also known as a box-and-whisker plot, is a graphical representation of the distribution of numerical data. It displays the five-number summary of a dataset, which includes the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value.

Here,

From the box plots, we can make the following observations:

A) The range for Mrs. Morales's class appears to be 100-50 = 50.

The range for Mr. Florez's class appears to be 95-55 = 40.

Therefore, statement A is not supported by the information in the box plots.

B) The box plot for Mr. Flores's class is more symmetrical than the box plot for Mrs. Morales's class, which is skewed to the left.

Therefore, statement B is supported by the information in the box plots.

C) The median score for Mrs. Morales's class appears to be between 70 and 75.

The median score for Mr. Florez's class appears to be between 75 and 80.

Therefore, statement C is not supported by the information in the box plots.

D) The interquartile range for Mrs. Morales's class appears to be between 60 and 80, which is a range of 20.

The interquartile range for Mr. Florez's class appears to be between 65 and 85, which is also a range of 20.

Therefore, statement D is supported by the information in the box plots.

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Therefore , the solution of the given problem of equation comes out to be required equation is 0.05b = 4.80 .

Mathematical formulas frequently use the same variable letter to try to impose unity between two assertions. Many academic numbers are shown to be equal using mathematical equations, also known as assertions. Using y + 6 as an illustration, the normalise does not divide 12 into two parts, but instead b + 6. It is possible to determine the connection integer between each sign part and the number of lines. The significance of a symbol usually contradicts itself.

Here,

The following equation can be set up to represent the circumstance if x is the procedure's cost:

=>  5% of x = $4.80

The percentage must first be divided by 100 to become a decimal before we can answer for x:

=>  0.05x = $4.80

Then, by dividing both lines by 0.05, we can separate out x:

=>  x = $96

Consequently, the process will cost you $96. It is possible to confirm that the gratuity is 5% of $96:

=>  5% of $96 = 0.05 × $96 = $4.80

So, the solution is:

=> 0.05b = 4.80

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The intersection set, A∩ B, of the sets A = {2, 5, 6, 14, 16} and B = {1, 3, 6, 8, 14 } is equals to the { 6, 14}.

For any two sets A and B, the intersection, A∩ B (read as A intersection B) lists all the elements that are present in both sets, and are the common elements of A and B. The intersection of sets is denoted by the symbol '∩'. It is subset of both of sets, A and B. For example, if Set A = {1,2,3,4,5} and Set B

= {3,4,6,8}, A ∩ B = {3,4}.

Here, we have two sets A = {2, 5, 6, 14, 16} and B = {1, 3, 6, 8, 14 }

The common elements in sets A and B

= { 6,14 }

Therefore, the intersection of A and B,

A ∩ B = { 6, 14}.

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Complete question:

What is the intersection of the sets A = {2, 5, 6, 14, 16} and B = {1, 3, 6, 8, 14}?

The exact values are sin 195 = 0.9763, cos 165 = 0.6088, cos θ/2 = 0.8944, and cot θ/2 = -0.2764.

To find the exact value of sin 195 and cos 165 using a half-angle identity, we need to use the following formulas:

sin(x/2) = √[(1-cosx)/2] and cos(x/2) = √[(1+cosx)/2]

For sin 195, we can use the half-angle identity for sine:

sin(195/2) = sin(97.5) = √[(1-cos195)/2] = √[(1+0.9063)/2] = √(0.9532) = 0.9763

For cos 165, we can use the half-angle identity for cosine:

cos(165/2) = cos(82.5) = √[(1+cos165)/2] = √[(1-0.2588)/2] = √(0.3706) = 0.6088

For question 8, we can use the given values and the half-angle identities to find the exact values of cos θ/2 and cot θ/2.

For cos θ/2 given sin θ = -4/5 with 180º < θ < 270º, we can use the half-angle identity for cosine:

cos(θ/2) = √[(1+cosθ)/2] = √[(1+√(1-sin^2θ))/2] = √[(1+√(1-(-4/5)^2))/2] = √[(1+√(1-(16/25)))/2] = √[(1+√(9/25))/2] = √[(1+(3/5))/2] = √(0.8) = 0.8944

For cot θ/2 given tan θ = -√5/2 with 90° < θ < 180°, we can use the half-angle identity for cotangent:

cot(θ/2) = (1+cosθ)/sinθ = (1+√(1-sin^2θ))/sinθ = (1+√(1-(1/tan^2θ)))/(1/tanθ) = (1+√(1-(1/(-√5/2)^2)))/(-√5/2) = (1+√(1-(4/5)))/(-√5/2) = (1+√(1/5))/(-√5/2) = (1+(1/√5))/(-√5/2) = (1-(1/√5))/(-√5/2) = -0.2764

Therefore, the exact values are sin 195 = 0.9763, cos 165 = 0.6088, cos θ/2 = 0.8944, and cot θ/2 = -0.2764.

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The value of image of coordinates of triangle ABC are,

A' = (4, - 4)

B' = (5 - 1)

C' = (1, - 3)

And, After dilate DEFG by a scale factor of 3 about (- 5, 5) is,

D' = (- 15, 6)

E' = (- 12, 12)

F'  = (- 6, 12)

G' = (- 9, 6)

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

Given that;

Reflect the triangle ABC across the line y = x

And, Dilate DEFG by a scale factor of 3 about (- 5, 5).

Now,

We know that;

The rule of reflection across the line y = x is,

(x, y) → (y, x)

Thus, The value of coordinates of triangle ABC are,

A = (- 4, 4)

B = (- 1, 5)

C = (- 3, 1)

Hence, The image of coordinates of triangle ABC are,

A' = (4, - 4)

B' = (5 - 1)

C' = (1, - 3)

And, The coordinates of DEFG are,

D = (- 5, 2)

E = (- 4, 4)

F = (- 2, 4)

G = (- 3, 2)

Hence, After dilate DEFG by a scale factor of 3 about (- 5, 5) is,

D' = (- 5x3, 2x3) = (- 15, 6)

E' = (- 4x3, 4x3) = (- 12, 12)

F' = (- 2x3, 4x3) = (- 6, 12)

G' = (- 3x3, 2x3) = (- 9, 6)

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The vectors are linearly independent precisely when k≠a,b, where a=-18 and b=1/3.

If k is a real number, then the vectors (1,k), (k,3k+18) are linearly independent precisely when k≠a,b, where a=-18 and b=1/3. This is because for the vectors to be linearly independent, the equation c1(1,k) + c2(k,3k+18) = (0,0) should only have the trivial solution of c1 = c2 = 0. If k = a or k = b, then there will be nontrivial solutions for c1 and c2, making the vectors linearly dependent.

To find the values of a and b, we can set the equations for the x and y components equal to 0 and solve for k:

c1 + c2k = 0

c1k + c2(3k+18) = 0

From the first equation, we can solve for c1 in terms of c2 and k:

c1 = -c2k

Substituting this into the second equation:

-c2k^2 + c2(3k+18) = 0

c2(-k^2 + 3k + 18) = 0

Since c2 cannot be 0, we can divide both sides by c2 and solve the quadratic equation for k:

-k^2 + 3k + 18 = 0

(k-1/3)(k+18) = 0

So k = 1/3 or k = -18. Therefore, the vectors are linearly independent precisely when k≠a,b, where a=-18 and b=1/3.

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Answer:

$45072

Step-by-step explanation:

I = Prt

I = 36000(6.3/100)(4)

I = 9072

ending balance = 36000 + 9072

= 45072

Answer:

The formula to calculate simple interest is:

I = Prt

Where I is the interest, P is the principal amount borrowed, r is the interest rate per period, and t is the time period.

Using this formula, we can calculate the interest Ms. Lang will pay as follows:

I = Prt

I = 2300 * 0.13 * (30/12)

I = $975

Therefore, Ms. Lang will pay $975 in interest over the 30-month period.

To calculate the total amount she will repay, we need to add the interest to the principal amount borrowed:

Total amount = Principal + Interest

Total amount = $2300 + $975

Total amount = $3275

Therefore, Ms. Lang will repay a total of $3275, which includes the $2300 principal and $975 interest.

now, we're assuming this is on a simple interest rate, so hmm say a year has 12 months, so 30 months will just be 30/12 of a year.

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$2300\\ r=rate\to 13\%\to \frac{13}{100}\dotfill &0.13\\ t=years\to \frac{30}{12}\dotfill &\frac{5}{2} \end{cases} \\\\\\ I = (2300)(0.13)(\frac{5}{2}) \implies I = 747.5~\hfill \underset{ \textit{amount to be repaid} }{\stackrel{ 2300~~ + ~~747.5 }{\text{\LARGE 3047.5}}}[/tex]

Where are the questions?

Answer:

(24 divided by 3) + x

Step-by-step explanation:

i did it

Based on the histograms, it appears that Machine B is doing a better job of dispensing the correct amount of liquid.

The most common graph for displaying frequency distributions is a histogram.

To begin, Machine B's histogram is more symmetrically distributed around the 5.5-ounce mark, which is the target fill level.

Machine A's histogram, on the other hand, is skewed to the right, indicating that it is more likely to dispense more than 5.5 ounces.

Second, Machine B's histogram has a narrower spread, with most measurements falling between 5.4 and 5.6 ounces.

This indicates that the machine fills the bottles consistently within a narrow range of the target fill level.

Thus, the histogram for Machine A, on the other hand, has a wider range, with some measurements falling below 5.4 ounces and others exceeding 5.6 ounces, indicating a less consistent performance.

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Your question seems incomplete, the probable complete question is:

For the given function we have:

domain  [0, 76,867] and range [$0, $12,375,587]

We know that x represents the number of people in atendance, then:

R(x) = 161*x

Represents the revenue.

The domain is the set of possible inputs, and it will go between 0 and 76,867 (total number of seats).

The range will go between:

R(0) = 161*0 = 0

R(76,867) = 76,867*161 = 12,375,587

Then the domain is all the integers in [0, 76,867] and the range is [$0, $12,375,587]

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Martina can run 4,920 more feet this year compared to last year.

Martina can run 3 miles without stopping. Last year she could run 3,640 yards without stopping. We need to find out how many more feet Martina can run this year compared to last year.

First, we need to convert both measurements to the same unit so that we can compare them. We will convert both measurements to feet.

1 mile = 5,280 feet
1 yard = 3 feet

So, 3 miles = 3 x 5,280 feet = 15,840 feet

And, 3,640 yards = 3,640 x 3 feet = 10,920 feet

Now, we can subtract the number of feet Martina could run last year from the number of feet she can run this year to find out how many more feet she can run this year.

15,840 feet - 10,920 feet = 4,920 feet

Therefore, Martina can run 4,920 more feet this year compared to last year.

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The relative maxima.

To find the relative maxima and minima of the function f(x) = x + 9/x + 5, we need to first find the derivative of the function.

The derivative of f(x) is f'(x) = 1 - 9/x^2.

Next, we need to set the derivative equal to zero and solve for x to find the critical points.

1 - 9/x^2 = 0

9/x^2 = 1

x^2 = 9

x = ±3

So, the critical points are x = 3 and x = -3.

To determine if these are relative maxima or minima, we need to use the second derivative test.

The second derivative of f(x) is f''(x) = 18/x^3.

Plugging in x = 3, we get f''(3) = 18/27 = 2/3, which is positive. This means that x = 3 is a relative minima.

Plugging in x = -3, we get f''(-3) = 18/-27 = -2/3, which is negative. This means that x = -3 is a relative maxima.

Therefore, the relative maxima is at x = -3 and the relative minima is at x = 3.

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a) The sum of their times is of 4x + 3.

b) The property used is the combination of like terms, which we use to add the terms with the variable x.

c) If Doug ran the race in 27 minutes, the total time for the team is of 111 minutes.

Doug finished first, and the times of the other two athletes are as function of Doug, hence the variable x represents the time needed for Doug to finish the race, that is:

D = x.

Kim finished 3 minutes after Doug, hence:

K = D + 3

K = x + 3.

Conner finished with a time that was twice Doug’s time, hence:

C = 2x.

Then the total time is given as follows:

T = D + K + C

T = x + x + 3 + 2x

T = 4x + 3.

If Doug ran the race in 27 minutes, the total time for the team is obtained as follows:

T = 4 x 27 + 3

T = 111 minutes.

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There will be about 111.25 square feet of space remaining in the room after the ping pong table is put in.

First, we need to convert the scale of the blueprint to a real-world scale. Since the scale is 1.5 in 5 feet, we can write this as:

1.5 inches : 5 feet

To convert this to a ratio that we can use, we can divide both sides by 1.5 inches:

1 inch : 3.33 feet

Now we can use this ratio to convert the dimensions of the room from the blueprint to real-world dimensions:

3 inches x 3.33 feet/inch = 9.99 feet

4.8 inches x 3.33 feet/inch = 15.96 feet

So the room is approximately 10 feet by 16 feet.

Next, we need to see if the ping pong table will fit in the room. The table is 9 feet by 5 feet, which means it needs at least a space of 9 feet x 5 feet = 45 square feet. To convert this to the scale of the blueprint, we can use the inverse of the ratio we calculated earlier:

1 foot : 1/3.33 inches

So the table needs at least a space of 9 feet x (1/3.33 inches/foot) = 27 inches by 5 feet x (1/3.33 inches/foot) = 15 inches on the blueprint.

The space in the room is 3 inches x 4.8 inches on the blueprint, which is equivalent to 3 inches x (1/1.5 inches/foot) = 2 feet and 4.8 inches x (1/1.5 inches/foot) = 1 foot 7.8 inches in real-world dimensions.

Since the table will fit on the blueprint, we can now calculate the remaining space in the room. The total area of the room is approximately 10 feet x 16 feet = 160 square feet.

The area taken up by the ping pong table is approximately 9 feet x 5 feet = 45 square feet, or 27 inches x 15 inches on the blueprint. To convert this to square feet, we can use the ratio we calculated earlier:

27 inches x (1/12 feet/inch) x 15 inches x (1/12 feet/inch) = 3.75 square feet

So the remaining space in the room is approximately 160 square feet - 45 square feet = 115 square feet - 3.75 square feet

= 111.25 square feet.

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Answer:

1. What is the maximum length of a chess table that is still within the tolerance limit?

To find the maximum length of a chess table that is still within the tolerance limit, we add the tolerance to the average length:

110 cm + 16.5 cm = 126.5 cm

Therefore, the maximum length of a chess table that is still within the tolerance limit is 126.5 cm.

2. What is the minimum length of a chess table that is still within the tolerance limit?

To find the minimum length of a chess table that is still within the tolerance limit, we subtract the tolerance from the average length:

110 cm - 16.5 cm = 93.5 cm

Therefore, the minimum length of a chess table that is still within the tolerance limit is 93.5 cm.

Step-by-step explanation:

Answer:

64.15°

Step-by-step explanation:

here

theta = 64.15°

( solution in picture)

The area of the pond after rounding to the nearest hundredth is 39 feet²

An area is total space occupied by two-dimensional or flat surfaces. In other words we can say that it is a number of unit squares present in a closed figure. We use various units for measurement of area like, cm², m², ft², mm².

To find the area of a semicircle, we need to first find the radius of the pond.

Let's assume the diameter of the pond is 10 feet.

Since a semicircle is half of a circle, the radius of the pond is half the diameter, which is 5 feet.

Now we can use the formula for the area of a semicircle, which is:

A = (1/2)πr²

where A is the area and r is the radius.

Plugging in the values, we get:

A = (1/2) × 3.14 × 5²

A = 39.25

Rounding to the nearest whole number, the area of the pond is approximately 39 square feet.

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The surface areas of these cylinders are 75.39 cm, 835.76 cm,  1324.11 cm, and 3543.71 cm.

The surface area of a cylinder can be calculated by using the formula  2π r h + 2π r², which means we need the radius and the height to calculate the area. Let's now use this formula to know the area of each cylinder:

Cylinder 1:

2π 2 4+ 2π 2² = 75.39 cm

Cylinder 2:

2π 7 12+ 2π 7² = 835.76 cm

Cylinder 3:

2π 8.2  17.5+ 2π 8.2² = 1324.11 cm

Cylinder 4:

2π 12 35 + 2π 12r² = 3543.71 cm

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Local Minima is 5 and 80

The critical numbers, a and b, of f'(x)=x^4−19x^3+135x^2+2 can be found by setting the derivative f'(x) = 0 and solving for x.

The derivative is: f'(x) = 4x^3 − 57x^2 + 270x

By setting the derivative equal to 0, we get:
4x^3 − 57x^2 + 270x = 0

Solving for x, we get:
x = 0, x = 5, x = 9

The critical numbers of f'(x) are 0, 5 and 9. The second derivative f''(x) is undefined at x = 0, so this is a local maximum point, while x = 5 and 9 are local minimum points.

Table:
Critical number | Second derivative (f''(x)) |  Value
0 | Undefined | Local Maximum
5 | -80 | Local Minimum
9 | 80 | Local Minimum

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Measure of ∟A is 119°

To solve for x, we must use the fact that the angles have the same measure, so ∟A = ∟B.

Substitute 7x + 24° for ∟A and 3x + 92° for ∟B and set them equal to each other:

7x + 24° = 3x + 92°

Subtract 3x from both sides:

7x - 3x = 92° - 24°

4x = 68°

Divide both sides by 4 to solve for x:

x = 17°

Now that we have x, we can substitute 17° for x into the original equation for ∟A to find its measure:

∟A = 7x + 24°

∟A = 7(17°) + 24°

∟A = 119°

Therefore, the measure of ∟A is 119°.

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From the given information provided, the number of students who brought a packed lunch and did not visit the gift shop is 32.

The number of students who brought a packed lunch and did not visit the gift shop can be calculated by subtracting the number of students who visited the gift shop from the number of students who bought a packed lunch:

32 - 0 = 32

A frequency tree is a visual representation of data that shows how the total number of individuals or items is distributed among different categories or groups. It is commonly used in statistics and probability to organize and analyze data in a hierarchical structure.

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Answer:

Keith bought 14 gallons of paint, and each classroom requires 7/4 gallons of paint.

To find how many classrooms can be painted, we can divide the total amount of paint by the amount of paint needed per classroom:

14 ÷ (7/4) = 8

Therefore, Keith can paint 8 classrooms in total.

Answer: 709 milliliters

Step-by-step explanation:

It is just the sum of the numbers so,

236 + 473 = 709

709 milliliters

Hope this helps!

Answer:

709 millimeters

Step-by-step explanation:

Add it up 236 = 473 = 709

Answer: Option B is the correct answer

Step-by-step explanation:

Area of lateral surface = 3 x Area of rectangle

The rectangle is of  5 inches wide and 14 inches long.

Area of rectangle =  5 x 14 = 70 square inches

Area of lateral surface = 3 x Area of rectangle

Area of lateral surface = 3 x 70 = 210 square inches

Option B is the correct answer.