what number is 8.2% of 50

Final answer:

The two-digit number where the tens digit is three more than the units digit and the number is seven times the sum of its digits is 74.

Explanation:

The question involves finding a two-digit number that fits two conditions: it is seven times the sum of its digits, and its tens digit is three more than the units digit. To solve this, we set up the following equations. Let x represent the tens digit and y represent the units digit.

Substituting x from the second equation into the first equation, we get 10(y + 3) + y = 7(y + 3 + y). Solving this, we find y = 4 and therefore x = 4 + 3, which gives x = 7. Thus, the number is 74.

Answer:

The required equation is [tex]C=\frac{5}{9}(F-32)[/tex]

Step-by-step explanation:

Consider the provided equation.  

[tex]F= \frac{9}{5}C+32[/tex]

Solve the formula for C.

Subtract 32 from both sides.

[tex]F-32= \frac{9}{5}C+32-32[/tex]

[tex]F-32= \frac{9}{5}C[/tex]

Multiply both the sides by 5/9.

[tex]\frac{5}{9}(F-32)= \frac{9}{5}C\times \frac{5}{9}[/tex]

[tex]C=\frac{5}{9}(F-32)[/tex]

Hence the required equation is [tex]C=\frac{5}{9}(F-32)[/tex]

The formula to convert Fahrenheit to Celsius is C = 5/9( F - 32 ).

Given the equation in the question:

F = (9/5)C + 32

To solve for C (Celsius) in the conversion formula F = (9/5)C + 32, first, isolate the terms that contain C on one side of the equation.

F = (9/5)C + 32

Subtract 32 from both sides of the equation:

F - 32 = (9/5)C + 32 - 32

F - 32 = (9/5)C

Next, multiply both sides by the reciprocal of the coefficient of C: 5/9:

5/9( F - 32 ) = 5/9 × (9/5)C

5/9( F - 32 ) = C

C = 5/9( F - 32 )

Therefore, C equals C = 5/9( F - 32 ).

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Answer:

The exact value is [tex]\cfrac{\sqrt{3}}3[/tex]

Step-by-step explanation:

Since 60 degrees is an angle we can find on the unit circle, the goal to get an exact value is to use the elements of the unit circle, which are exact values of sine and cosine.

Writing cotangent in terms of sine and cosine

We can use the trigonometric identity

[tex]\cot \theta = \cfrac{\cos \theta }{\sin \theta }[/tex]

Thus for the exercise we will have

[tex]\cot 60^\circ = \cfrac{\cos 60^\circ }{\sin 60^\circ }[/tex]

Identifying the known exact values.

From the unit circle that you can see on the attached image below, we have to identify the exact values of cosine and sine of 60  degrees.

So first try to look for the angle 60 degrees, there you will see a point that has a pair of values,  those represent (cosine, sine), thus we get:

[tex]\cos 60^\circ=\cfrac 12 \\\\\sin 60^\circ = \cfrac{\sqrt3}2[/tex]

Finding the exact value of cot 60 degrees.

We can replace the exact values of sine and cosine on the trigonometric identity for cotangent.

[tex]\cot 60^\circ = \cfrac{\cfrac 12 }{\cfrac{\sqrt 3}2 }[/tex]

Working with the reciprocal we get

[tex]\cot 60^\circ = \cfrac 12\times \cfrac2{\sqrt 3}[/tex]

Simplifying we get

[tex]\cot 60^\circ = \cfrac 1{\sqrt 3}[/tex]

Rationalizing since we usually do not want square roots on the denominator we get

[tex]\cot 60^\circ = \cfrac 1{\sqrt 3} \times \cfrac{\sqrt 3}{\sqrt 3}\\\boxed{\cot 60^\circ = \cfrac {\sqrt 3}3}[/tex]

And that is the exact value of cotangent of 60 degrees.

The exact value of cot(60°) is √3 / 3.

To find the exact value of cot(60°), we can use the identity:

cot(θ) = 1 / tan(θ)

Since tan(θ) = sin(θ) / cos(θ), we need to find the values of sin(60°) and cos(60°).

In a 30-60-90 degree triangle, the sides are in the ratio 1 : √3 : 2. Since the angle is 60°, the opposite side (opposite the 60° angle) has length √3 and the adjacent side (adjacent to the 60° angle) has length 1.

Using these values, we can calculate the sine and cosine of 60°:

sin(60°) = opposite/hypotenuse = √3/2

cos(60°) = adjacent/hypotenuse = 1/2

Now, we can find cot(60°):

cot(60°) = 1 / tan(60°)

= 1 / (sin(60°) / cos(60°))

= 1 / (√3/2 / 1/2)

= 1 / (√3/1)

= 1 / √3

= √3 / 3

Therefore, the exact value of cot(60°) is √3 / 3.

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To calculate the sum of 12x² + 9x², add the coefficients (12 and 9) to get 21 and then multiply by x², resulting in a sum of 21x².

To find the sum of the expression 12x² + 9x², you simply add the like terms. Both terms have an x² component, so they can be combined.

Here's how you do it:

First, identify the coefficients of the x² terms, which are 12 and 9.

Next, add these two coefficients together: 12 + 9 = 21.

Finally, multiply the sum of the coefficients by x2 to get the final answer: 21x².

Therefore, the sum of 12x² + 9x² is 21x².

a. Probability density function (pdf) of X: 0.247 for 0.20 < x < 4.25.

b. Probability that diameter exceeds 2 mm: 0.556.

c. Probability that diameter is within 2 mm of the mean diameter: 0.988.

(a) Probability density function (pdf) of X:

For a uniform distribution with A = 0.20 and B = 4.25, the pdf is constant within the range A to B and 0 elsewhere. Therefore:

f(x) = 1 / (B - A) = 1 / (4.25 - 0.20) = 1 / 4.05 ≈ 0.247 for 0.20 < x < 4.25

(b) Probability that diameter exceeds 2 mm:

P(X > 2) = (4.25 - 2) * f(x) = 2.25 * 0.247 ≈ 0.556

(c) Probability that diameter is within 2 mm of the mean diameter:

The mean of a uniform distribution is (A + B)/2 = (0.20 + 4.25)/2 = 2.225.

So, we need to find P(2.225 - 2 < X < 2.225 + 2), which is P(0.225 < X < 4.225).

P(0.225 < X < 4.225) = (4.225 - 0.225) * f(x) = 4 * 0.247 ≈ 0.988

Complete question:

An article considered the use of a uniform distribution with

A = 0.20 and B = 4.25

for the diameter X of a certain type of weld (mm).

(a) Determine the pdf of X. (Round your answers to three decimal places.)

0.2<x<4.25

(b) What is the probability that diameter exceeds 2 mm? (Round your answer to three decimal places.)

(c) What is the probability that diameter is within 2 mm of the mean diameter? (Round your answer to three decimal places.)

The approximate attendance at the movie theater in nine years, with a continuous growth rate of 4.2%, is around 79,918.

The question provided can be solved using the formula for continuous growth, A = P ert, where A is the final amount, P is the initial principal amount (56,000 in this case), r is the rate of growth (4.2% or 0.042 when expressed as a decimal), and t is time in years (9 years here).

To find the approximate attendance at the movie theater in nine years, we substitute our values into the formula: A = 56000 x e(0.042 x 9). After calculating this, we find that the approximate attendance at the movie theater after nine years is about 79,918.

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Answer:

A. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.

C. The ball reaches its maximum height about 4 seconds after it is thrown

Step-by-step explanation:    • The ball is at the same height as the building between 8 and 10 seconds after it is thrown. TRUE - the height is zero somewhere in that interval, hence the ball is the same height from which it was thrown, the height of the roof of the building.

• The height of the ball decreases and then increases. FALSE - at t=2, the height is greater than at t=0.

• The ball reaches its maximum height about 4 seconds after it is thrown. TRUE - the largest number in the table corresponds to t=4.

• The ball hits the ground between 8 and 10 seconds after it is thrown. FALSE - see statement 1.

• The height of the building is 81.6 meters. FALSE - the maximum height above the building is 81.6 meters. Since the ball continues its travel to a distance 225.6 meters below the roof of the building, the building is at least that high.

To find the initial number of chocolates Jenny had, we solve the equation (x - 2)/2 = 6, which results in x = 14. Therefore, Jenny had 14 chocolates initially.

The question asks to determine how many chocolates Jenny had in the beginning if she eats two and gives half of the remainder to Lisa, who ends up with six chocolates. To solve this, we let x represent the initial number of chocolates Jenny had. After eating two chocolates, Jenny has x - 2 left. She gives half of this remainder to Lisa, which means Lisa receives (x - 2)/2 chocolates. Since Lisa has six chocolates, we set up the equation (x - 2)/2 = 6. Solving this gives us x - 2 = 12 and therefore, x = 14. Hence, Jenny had 14 chocolates in the beginning. The correct answer is option C. 14.

Final answer:

△RST is a right triangle because the slope of line RS is 3/2 and the slope of line RT is 0, indicating that RS is perpendicular to RT.

Explanation:

To determine whether △RST is a right triangle, we can calculate the slopes of the sides to check for perpendicularity. A right triangle will have one pair of sides that are perpendicular to each other, meaning their slopes will be negative reciprocals.

The slope of line RS is calculated using the coordinates R(-3,1) and S(-1,4) as:

Slope of RS = (4 - 1) / (-1 + 3) = 3 / 2

The slope of line RT is calculated using the coordinates R(-3,1) and T(3,1) as:

Slope of RT = (1 - 1) / (3 + 3) = 0 / 6 = 0

Since the slope of RS is a non-zero finite number and the slope of RT is zero, they are perpendicular to each other because the slope of a line perpendicular to a horizontal line (slope of 0) is undefined, which is the negative reciprocal of 0.

Therefore, the correct statement is:

△RST is a right triangle because RS‾ is perpendicular to RT‾.

Final answer:

Upon calculating the slopes of the sides of △RST, it is concluded that none of the sides are perpendicular to one another, which means that △RST is not a right triangle.

Explanation:

To determine if △RST is a right triangle, we need to calculate the slopes of the sides to check for perpendicularity because perpendicular lines have slopes that are negative reciprocals of each other. Let's calculate the slopes of line segments RS, ST, and RT.

Slope of RS is given by (4 - 1)/(-1 + 3) = 3/2.
Slope of ST is (4 - 1)/(-1 - 3) = 3/-4 = -3/4.

Slope of RT is (1 - 1)/(3 + 3) = 0/6 = 0.

Since the slope of RT is 0, it means that RT is a horizontal line. The slope of RS is 3/2, and the slope of ST is -3/4. These slopes are not negative reciprocals of each other. Hence, none of the lines are perpendicular to each other, and we can conclude that △RST is not a right triangle because no two of its sides are perpendicular.

To find the cost of the love seat, couch, and chair, set up a system of equations and solve for the variables.

To find the cost of the love seat, the couch, and the chair, we need to set up a system of equations based on the given information. Let's represent the cost of the chair as x. Since the couch costs twice as much as the chair, its cost will be 2x. The love seat costs $400 more than the couch, so its cost will be 2x + $400. The sum of the costs of all three pieces of furniture is $1565. Using these equations, we can solve for x, and then find the costs of the love seat, the couch, and the chair.

Equations:

Cost of the Chair: $233

Cost of the Couch: 2x = 2($233) = $466

Cost of the Love Seat: 2x + $400 = 2($233) + $400 = $466 + $400 = $866

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Answer:

The equation that models this situation : [tex]4\times (6+x)=48[/tex]

Duration of bicycle rented on Sunday was of 6 hours.

Step-by-step explanation:

Cost of per hour to rent the bicycle = $4

Duration of bicycle rented on Saturday = 6 hours

Duration of bicycle rented on Sunday = x

Total mount paid = $48

[tex]\$4\times (6+x)=$48[/tex]

[tex]4\times (6+x)=48[/tex]

For solving for x:

[tex]x=\frac{48}{4}-6= 6[/tex]

Duration of bicycle rented on Sunday was of 6 hours.

The amount of water flowing each second is:

rate = 5.2 x 10^5 gallons / second

Since we know that:

1 hour = 3600 seconds

Therefore:

rate = (5.2 x 10^5 gallons / second) * (3600 seconds / hour)

rate = 1.872 x 10^9 gallons / hour

Final answer:

The average volume of water that flows over the falls in an hour is 1.89 x 10^9 gallons/hour.

Explanation:

To calculate the amount of water that flows over the falls in an hour, we need to convert the given flow rate from gallons per second to gallons per hour.

There are 3600 seconds in an hour, so we can multiply the flow rate (5.25 x 105 gallons per second) by 3600 to find the flow rate in gallons per hour.

The calculation would be: 5.25 x 105 gallons/s * 3600s/hour = 1.89 x 109 gallons/hour.

The result, in scientific notation, is 1.89 x 109 gallons/hour.

The equation that represents the cost y (in dollars) of renting a jet ski for x hours is y = -87.50x + 525.

The equation that represents the cost y (in dollars) of renting a jet ski for x hours can be found by analyzing the given information. Let's break it down step by step:

Therefore, the equation that represents the cost y (in dollars) of renting a jet ski for x hours is y = -87.50x + 525.

Answer: D (Neither option A nor option B will allow them to meet their goal.

Hope this helps!