There were 81 people sitting in a school auditorium of which 19 were teachers, while the rest were students. How many teachers were sitting in the auditorium?

The equation is y = x + 3 which shows the relationship of the two activities.

The equation to show the relationship between the number of hours Julie practices her violin each week (y) and the number of hours she studies (x) is:

y = x + 3

This means that the number of hours Julie practices her violin each week is equal to the number of hours she studies, plus three additional hours.

A linear equation is an equation that represents a straight line on a graph. It can be written in the form:

y = mx + b

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Answer:

Step-by-step explanation:

4/625 x 625/9 = 4 x 1 / 5 x 5 x 5 x 1 = 4/625. The cross cancellation did not change the result.

The amount of total and per-unit overhead allocated to each of the three products using overhead rate is equal to,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530   $1,060,000

Clarinets $795    $1,192,500

Oboes $397.5     $695,625

Total $1,722.5    $2,948,125

Budgeted factory overhead cost = $2,948,125

The single plantwide overhead rate

= Dividing the budgeted factory overhead cost by the total budgeted direct labor hours.

For this,

Flutes= 2,000×2

         = 4,000 hours

Clarinets= 1,500×3

              = 4,500 hours

Oboes= 1,750×1.5

          = 2,625 hours

Total direct labor hours = 11,125

Substitute the value we have,

⇒ Single plantwide overhead rate = $2,948,125 / (2,000 x 2.0 + 1,500 x 3.0 + 1,750 x 1.5)

= $2,948,125 / 11,125

= $265 per direct labor hour

To allocate overhead to each product

=Multiply the overhead rate by the budgeted direct labor hours per unit for each product.

Substitute the value we have,

Flutes,

$265 x 2.0 = $530 total overhead cost, $265 per unit

Clarinets,

$265 x 3.0 = $795 total overhead cost, $265 per unit

Oboes,

$265 x 1.5 = $397.5 total overhead cost, $265 per unit

And

Total factory overhead cost allocated = Estimated manufacturing overhead rate×  Actual amount of allocation base

For,

Flutes

= 4,000× 265

= $1,060,000

Clarinets

= 4,500×265

= $1,192,500

Oboes

= 2,625×265

= $695,625

This implies,

The total and per-unit overhead allocated to each product, rounded to the nearest dollar is,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530 $1,060,000

Clarinets $795 $1,192,500

Oboes $397.5  $695,625

Total $1,722.5 $2,948,125

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The above question is incomplete, the complete question is:

Bach Instruments Inc. makes three musical instruments: flutes, clarinets, and oboes. The budgeted factory overhead cost is $2,948,125. Overhead is allocated to the three products on the basis of direct labor hours. The products have the following budgeted production volume and direct labor hours per unit:

Budgeted Production Volume Direct Labor Hours Per Unit

Flutes 2,000 units 2.0

Clarinets 1,500 3.0

Oboes 1,750 1.5

a. Determine the single plantwide overhead rate.

$ per direct labor hour

b. Use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.

Total Per Unit

Factory Overhead Cost Factory Overhead Cost

Flutes $ $

Clarinets

Oboes

Total $

The difference between the means expressed as a multiple of the mean absolute deviation is 2.3.

To find the difference between the means expressed as a multiple of the mean absolute deviation, we need to calculate the absolute difference between the two means and divide it by the mean absolute deviation.

The absolute difference between the means is:

|30.2 - 25.6| = 4.6

To express this difference as a multiple of the mean absolute deviation, we divide it by the mean absolute deviation:

4.6 / 2 = 2.3

Therefore, the difference between the means expressed as a multiple of the mean absolute deviation is 2.3.

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The ladder reaches up 20ft the side of the house.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem can be used to solve problems involving right triangles, such as finding the length of the sides or the height of an object.

In this problem, we are given the length of the ladder and the height up the side of the house that it reaches.

We can use the Pythagorean theorem to find the distance from the base of the ladder to the side of the house.

We can then use this distance and the height up the side of the house that the ladder reaches to find the length of the ladder using the Pythagorean theorem again.

Let's call the distance from the base of the ladder to the side of the house "x". We can then use the Pythagorean theorem to find the height that the ladder reaches up the side of the house.

According to the Pythagorean theorem, the length of the ladder (which is the hypotenuse of the right triangle formed by the ladder, the ground, and the side of the house) is equal to the square root of the sum of the squares of the other two sides.

So, if we let "h" be the height up the side of the house that the ladder reaches, we have:

ladder length = √(x^2 + h^2)

We know that the ladder is 20ft long and reaches up 16ft, so we can set up the equation:

20 = √(x^2 + 16^2)

Squaring both sides of the equation, we get:

400 = x^2 + 256

Subtracting 256 from both sides, we get:

144 = x^2

Taking the square root of both sides, we get:

x = 12

So the ladder is leaning against the house 12ft away from the base, and we can use the Pythagorean theorem to find the height up the side of the house that the ladder reaches:

ladder length = √(12^2 + 16^2) = √(144 + 256) = √400 = 20

Therefore, the ladder reaches up 20ft the side of the house.

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Answer: 8

Step-by-step explanation:

okay so do 17 x 17 - 15 x 15

then to that result do :

squared root.

The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is undefined. This is because as x approaches 2, the denominator (x^2-4) approaches 0, which means that the fraction as a whole is undefined. Therefore, there is no value that the limit can approach.

The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is:
Step 1: Recognize that the given expression can be simplified. Notice that the denominator, x^2 - 4, is a difference of squares, so it can be factored as (x-2)(x+2).
Step 2: Simplify the expression by canceling the common factors in the numerator and the denominator: (x-2)(x+2) / (x-2)(x+2) simplifies to 1, because the factors (x-2)(x+2) cancel each other out.
Step 3: Now that the expression is simplified, substitute x = 2 to find the value of the limit: lim x->2 | 1 = 1.
Your answer: The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is 1.

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The surface area of the cone is approximately 793.10 cm².

What is cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

What is area of cone?

The surface area of a cone is given by the formula:

Surface Area = πr² + πrℓ

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

To find the surface area of a cone, we need to find the area of its base and the area of its lateral surface and then add them together.

The formula for the lateral surface area of a cone is given by:

L = πrℓ

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

The formula for the area of the base of a cone is given by:

B = πr²

where r is the radius of the base.

Given the slant height ℓ = 19 and the radius r = 9, we can find the lateral surface area of the cone as follows:

L = πrℓ

= π(9)(19)

≈ 538.63 cm²

Next, we can find the area of the base of the cone as follows:

B = πr²

= π(9)²

≈ 254.47 cm²

Therefore, the surface area of the cone is:

A = B + L

≈ 793.10 cm²

So, the surface area of the cone is approximately 793.10 cm².

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y=(x+3)^2-1

have a good day :)

Answer: F(x) = (x + 3)^2 - 1

Step-by-step explanation:

The probability of drawing a star-shaped chip is 5/8.

The theoretical probability of drawing a star-shaped circular chip from the bag is 5/8 or 0.625 or 62.5%. Out of the total of eight circular chips, five are stars, and three are squares.

Therefore, the probability of drawing a star-shaped chip is the ratio of the number of star-shaped chips to the total number of chips in the bag, which is 5/8.

To understand this conceptually, we can think of probability as a fraction where the numerator is the number of favorable outcomes (in this case, drawing a star-shaped chip) and the denominator is the total number of possible outcomes (all the circular chips in the bag).

Thus, the theoretical probability of drawing a star-shaped chip is 5/8 because there are five star-shaped chips out of the total eight circular chips in the bag.

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The mechanic used 412 - 411.375 = 0.625 gallons of motor oil during the day.

To solve this problem, we need to convert both measurements to the same unit.

1 gallon = 8 pints (since there are 8 pints in a gallon)

So the mechanic started with:

412 gallons * 8 pints/gallon = 3,296 pints

And ended with:

5 pints

To find how much motor oil the mechanic used during the day, we can subtract the ending amount from the starting amount:

3,296 pints - 5 pints = 3,291 pints

To convert this back to gallons, we divide by 8:

3,291 pints / 8 pints/gallon = 411.375 gallons (rounded to three decimal places)

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Answer:

no it is not possible

Step-by-step explanation:

The Stafford loan will have a lower balance by $272.54 at the time of repayment.

For the Stafford Loan, the principal is $10,720 and the interest rate is 5.95% compounded monthly, so the monthly interest rate is 0.0595/12 = 0.00495833.

After 12 months, the balance of the loan will be:

$10,720(1 + 0.00495833)¹² = $11,204.60

After the time of six-month grace period, the balance will be:

$11,204.60(1 + 0.00495833)⁶ = $11,689.66

For the PLUS Loan, the principal is $11,443.63 and the interest rate is 6.55% compounded monthly, so the monthly interest rate is 0.0655/12 = 0.00545833.

After 12 months, the balance of the loan will be:

$11,443.63(1 + 0.00545833)¹² = $11,962.20

Therefore, the Stafford Loan will have a lower balance at the time of repayment by:

$11,962.20 - $11,689.66 = $272.54

Hence, the Stafford loan will have a lower balance by $272.54 at the time of repayment.

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a) Para calcular este límite, podemos dividir tanto el numerador como el denominador por n² y luego aplicar la regla de L'Hôpital:

lim n → ∞ [(6 - 4n²)/(2n²)]

= lim n → ∞ [6/(2n²) - (4n²)/(2n²)]

= lim n → ∞ [3/n² - 2]

= -2

Por lo tanto, el límite es -2.

b) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:

lim n → ∞ [(4n² + 3n - 2)/(2n³ - 4n)]

= lim n → ∞ [(4/n + 3/n² - 2/n³)/(2/n² - 4/n²)]

= lim n → ∞ [(4 + 3/n - 2/n²)/(2 - 4/n)]

= lim n → ∞ [(4n + 3 - 2n²)/(2n² - 4)]

= lim n → ∞ [-2n²/(2n² - 4)]

= -1

Por lo tanto, el límite es -1.

c) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:

lim n → ∞ [(2n³ - 4n)/(4n)]

= lim n → ∞ [(2n² - 4)/(4)]

= lim n → ∞ [(n² - 2)/2]

= +∞

Por lo tanto, el límite es +∞.

d) Podemos dividir tanto el numerador como el denominador por x⁴ para simplificar el límite:

lim x → ∞ [-8x⁴ + 2]/[2x² + 4]

= lim x → ∞ [-8 + 2/x⁴]/[2/x² + 4/x⁴]

= -4/1

= -4

Por lo tanto, el límite es -4.

The solution to the differential equation with the given initial condition is:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

To solve equation, we can separate the variables and write:

[tex]1/e^{(2u)} du = e^{(9t)} dt[/tex]

Integrating both sides, we get:

[tex]\int 1/e^{(2u)} du = \int e^{(9t)} dt[/tex]

Integrating the left side requires the substitution v = 2u, dv/du = 2, and du = dv/2, which gives:

[tex]\int 1/e^{(2u)} du = \int 1/2 * 1/e^v dv = -1/2 * e^{(-2u)}[/tex]

Integrating the right side gives:

[tex]\int e^{(9t)} dt = 1/9 * e^{(9t)}[/tex]

Substituting these integrals back into the original equation, we get:

[tex]-1/2 * e^{(-2u)} = 1/9 * e^{(9t)} + C[/tex]

where C is the constant of integration.

We can solve for the constant of integration using the initial condition u(0) = 4:

[tex]-1/2 * e^{(-24)} = 1/9 * e^{(90)} + C[/tex]

[tex]-1/2 * e^{(-8)} = 1/9 + C[/tex]

[tex]C = -1/2 * e^{(-8)} - 1/9[/tex]

Therefore, the solution to the differential equation [tex]du/dt = e^{(2u+9t)}[/tex] with initial condition u(0) = 4 is:

[tex]-1/2 * e^{(-2u)} = 1/9 * e^{(9t)} - 1/2 * e^{(-8)} - 1/9[/tex]

Solving for u, we get:

[tex]e^{(-2u)} = -2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9[/tex]

Taking the natural logarithm of both sides, we get:

[tex]-2u = ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

Dividing by -2, we get:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

Therefore, the solution to the differential equation with the given initial condition is:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

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There are different ways a person can make up a story about a word problem. The square root of 81 - 7 is 2. Hence the story is given below.

The hypothetical story goes like this: Mia was trying to figure out the length of the missing side of a square garden bed. She knew that the total area of the garden bed was 81 square feet. She also knew that she had already planted in a section of the garden bed that was 7 feet wide.

So, what will be the length of the missing side of the garden bed if Mia wants to make sure it is completely filled with soil. In order to solve this problem, Mia used the equation: the square root of 81-7=2. She plugged in the values she knew and solved for the missing side length. Therefore, The solution will be:= ✓81 - 7= 9 - 7= 2

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A story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet² and the width was 7 feet wide?

From the information, we are seeking a story problem that has the equation had the square root of 81 - 7 = 2.

This will be written as story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet ² and the width was 7 feet wide?

The length will be calculated thus :

= √ 81 - 7

= 9 - 7

= 2

The length of the missing side is 2 feet.

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Answer:

The order from smallest to largest price per package is: December, April, February.

Step-by-step explanation:

For the sale in February:

- Price per package = $4.49 ÷ 3 = $1.50 per package

For the sale in April:

- Price per package = $7.39 ÷ 5 = $1.48 per package

For the sale in December:

- Price per package = $5.88 ÷ 4 = $1.47 per package

If he drives at an average speed of 70 km/h for 45 min, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

To find the distance of Mr. Linden's journey from his home to his office, we can use the formula:

Distance = Speed x Time

Since Mr. Linden drives at an average speed of 70 km/h for 45 minutes, we first need to convert the time to hours:

45 minutes = 0.75 hours

Now, we can plug in the values we have into the formula:

Distance = 70 km/h x 0.75 hours
Distance = 52.5 km

Therefore, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

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The maximum rate of change occurs in the direction of this unit vector.

(a) To find the maximum rate of change of f at point P(1,0), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.

The gradient of f is:

∇f(x,y) = <y cos(xy), x cos(xy)>

At point P(1,0), we have:

∇f(1,0) = <0, cos(0)> = <0, 1>

The magnitude of the gradient is:

||∇f(1,0)|| = sqrt([tex]0^2[/tex] +[tex]1^2[/tex]) = 1

Therefore, the maximum rate of change of f at point P is 1, and it occurs in the direction of the unit vector in the direction of the gradient:

u = <0, 1>/1 = <0, 1>

So the maximum rate of change occurs in the y-direction.

(b) To find the maximum rate of change of f at point P(8,1.3), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.

The gradient of f is:

∇f(x,y,z) = <2x, 2y, 2z>

At point P(8,1.3), we have:

∇f(8,1.3) = <16, 2.6, 2(1.3)> = <16, 2.6, 2.6>

The magnitude of the gradient is:

||∇f(8,1.3)|| = sqrt[tex](16^2 + 2.6^2 + 2.6^2)[/tex]= sqrt(275.56) ≈ 16.6

Therefore, the maximum rate of change of f at point P is approximately 16.6, and it occurs in the direction of the unit vector in the direction of the gradient:

u = <16, 2.6, 2.6>/16.6 ≈ <0.963, 0.157, 0.157>

So the maximum rate of change occurs in the direction of this unit vector.

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50% of the beads Nolan received were orange.

To find the percentage of beads that were orange, we need to divide the number of orange beads by the total number of beads and then multiply by 100.

Percentage of orange beads = (Number of orange beads / Total number of beads) * 100

In this case, Nolan received a total of 86 beads, and 43 of them were orange.

Percentage of orange beads = (43 / 86) * 100

Calculating this expression:

Percentage of orange beads = 0.5 * 100

Percentage of orange beads = 50%

Therefore, Out of total number of  beads Nolan received, 50% were orange.

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A restaurant meal cost $90 and there was a 6% meals tax on the price of the meal. The meal tax was $5.40 and the total price of the meal with the meal tax was $95.40.

Find the meal tax, you need to calculate 6% of the $90 meal cost.
Convert the percentage to a decimal by dividing by 100, so 6% = 0.06.
Multiply the meal cost by the decimal. In this case, $90 × 0.06 = $5.40.
So, the meal tax was $5.40.
Find the total price of the meal with the meal tax, you need to add the meal tax to the original cost of the meal.
Add the meal tax to the original cost, so $90 + $5.40 = $95.40.
The total price of the meal with the meal tax was $95.40.

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Part A: The coefficient of the 4th degree term in the Taylor polynomial for f(x) = sin(4x) centered at x = 0 is -1/3! = -1/6.

Part B: Using a 4th degree Taylor polynomial for sin(x) centered at x = 0, we can write g(x) = sin(0.8) ≈ P4(0.8), where P4(0.8) is the 4th degree Taylor polynomial for sin(x) evaluated at x = 0.8.

Evaluating P4(0.8) using the formula for the Taylor series coefficients of sin(x), we get P4(0.8) = 0.8 - 0.008 + 0.00004 - 0.0000014 ≈ 0.78333. This estimate is very close to 1 because sin(0.8) is close to 1, and the Taylor series for sin(x) converges very rapidly for values of x close to 0.

Part C: The series { (-1)n / (2n + 1) } has a partial sum S when x = 1. To find an interval |S - S5| = R5| for which the actual sum exists, we can use the alternating series test. The alternating series test states that if the terms of a series alternate in sign, decrease in absolute value, and approach zero, then the series converges.

Since the terms of the series { (-1)n / (2n + 1) } alternate in sign and decrease in absolute value, we know that the series converges. To find an interval |S - S5| = R5|, we can use the remainder formula for alternating series, which states that |Rn| ≤ a_n+1, where a_n+1 is the first neglected term in the series.

Since the terms of the series decrease in absolute value, we know that a_n+1 ≤ |a_n|. Therefore, we have |R5| ≤ |a6| = 1/7!, which means that the actual sum of the series exists in the interval S - 1/7! ≤ S5 ≤ S + 1/7!. Therefore, an interval for which the actual sum exists is [S - 1/7!, S + 1/7!].

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Check the picture below.

The measure of arc CAD is either 240 or 260.

Without additional information, it is not possible to determine the measure of arc CAD with certainty. The measure of an arc depends on the central angle that subtends it.

If the central angle is known, the measure of the arc can be calculated using the formula: measure of arc = (central angle / 360) x circumference of the circle. However, without knowing the central angle, we cannot determine the measure of arc CAD.

Therefore, we need to be provided with additional information such as the measure of another angle that is related to the central angle, or the length of a chord that subtends the arc in order to determine the central angle and the measure of arc CAD.

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Cos 165 in terms of sine and cosine of acute angle would give cos(165) = -(1 + √3) / (2√2).

To find the cosine of 165 degrees in terms of sine and cosine of an acute angle, we can use the cosine angle addition formula:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Since 120 degrees is in the second quadrant, the cosine is negative, and the sine is positive:

cos(120) = -cos(60) and sin(120) = sin(60)

cos(165) = -cos(60)cos(45) - sin(60)sin(45)

Now we can plug in the values of the trigonometric functions:

cos(165) = - (1/2) x (1/√2) - (√3/2) x (1/√2)

cos(165) = -(1 + √3) / (2√2)

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Answer: £326

Step-by-step explanation:

Step 1: Calculate the cost per jumper

To find out how much Ryan spent on each jumper, we divide the total cost by the number of jumpers.

[tex]\frac{130}{40} = 3.25[/tex]

This gives us a cost of £3.25 per jumper.

Step 2: Calculate the revenue from selling 80% of the jumpers

Ryan sells 80% of the 40 jumpers, so:

[tex]\text{0.8 x 40 = 32}[/tex]

So he sold 32 Jumpers.

He sells each jumper for £12:

[tex]\text{32 x 12 = 384}[/tex]

So his revenue from selling these jumpers is £384

Step 3: Calculate the revenue from selling the remaining jumpers on the Buy one get one half price offer

Ryan has 8 jumpers left after selling 80% of them. He puts these on a Buy one get one half price offer, which means that for every jumper sold at full price, he sells another one at half price.

This means that he sells 4 jumpers at full price (£12 each) and 4 jumpers at half price (£6 each).

His revenue from selling these jumpers is:

[tex]\text{(4 x 12) + (4 x 6) = 72}[/tex]

Step 4: Calculate the total revenue

Ryan's total revenue is the sum of the revenue from selling 80% of the jumpers and the revenue from selling the remaining jumpers on the Buy one get one half price offer.

This is:

[tex]\text{384 + 72 = 456}[/tex]

So Ryan's total revenue is £456

Step 5: Calculate the total cost

Ryan's total cost is the amount he spent on buying the jumpers, which is £130.

Step 6: Calculate the profit

Ryan's profit is the difference between his total revenue and his total cost:

[tex]\text{456 - 130 = 326}[/tex]

Therefore, Ryan makes a profit of £326.

The perimeter of the shape is 53.7 units

Perimeter is a math concept that measures the total length around the outside of a shape.

A theorem of circle geometry states that the tangent from a point on a circle are equal.

Therefore the base sides is calculated as

9.9 + 3.2

= 13.1

since the perimeter is the addition of all the sides then;

P = 13.1 + 21.9 + 18.7

P = 53.7

therefore the perimeter of the triangle is 53.7

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The fraction, David used 13/8 cups of blueberries for the pancakes.

Let's solve it step-by-step using the given information.

1. David has 3 1/2 cups of blueberries.
2. He uses 1/4 cup for a breakfast smoothie.
3. He uses 1/2 of the remaining blueberries for pancakes.

Step 1: Calculate the remaining blueberries after making the smoothie.
3 1/2 - 1/4 = (7/2) - (1/4)

To subtract the fractions, they need a common denominator, which in this case is 4.

(7/2) * (2/2) - (1/4) = (14/4) - (1/4) = 13/4 cups

Step 2: Calculate the amount of blueberries used for the pancakes.
David uses 1/2 of the remaining blueberries for the pancakes, so:

(13/4) * (1/2) = 13/8 cups

David uses 13/8 cups of blueberries for the pancakes.

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Yes, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

The slope ratio is the ratio of the vertical rise to the horizontal run of the ramp, and it is equivalent to the tangent of the angle of inclination of the ramp.

The tangent of 4.8 degrees is approximately 0.0084, which means that for every 1 unit of vertical rise, there is 0.0084 units of horizontal run. To convert this to a ratio, we can multiply both sides by 100 to get:

1 unit of rise : 100 x 0.0084 = 0.84 units of run

Simplifying this ratio by dividing both sides by 0.84, we get:

1 unit of rise : 1.19 units of run

which is equivalent to a slope ratio of 1:12 (since 12 = 1/0.084). Therefore, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

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But there seems to be some missing information in your question. Please provide more context or details so that I can assist you accurately.
Hi! I'd be happy to help you evaluate the indefinite integral. Based on the provided terms and information, it seems like you want to evaluate the following integral:

∫x * sin(e * cos(x)) dx
To solve this integral, we can use integration by parts, which is defined as:
∫u dv = u * v - ∫v du
Let's choose u = x and dv = sin(e * cos(x)) dx. Then, we need to find du and v:
du = dx
v = ∫sin(e * cos(x)) dx

Unfortunately, the integral for v does not have a simple closed-form expression. However, you can use numerical methods or software (like Wolfram Alpha) to approximate it. Once you have an approximation for v, you can plug it back into the integration by parts formula to obtain an approximation of the original integral:
∫x * sin(e * cos(x)) dx ≈ x * v - ∫v dx

Keep in mind that this is an indefinite integral, so don't forget to add the constant of integration, C, to your final answer.

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