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Anna35 [415]
3 years ago
14

A 10-foot ladder is leaning against a vertical wall. If the bottom of the ladder is being pulled away from the wall at the rate

of 9 feet per second, at what rate is the area of the triangle formed by the wall, the ground, and the ladder changing, in square feet per second, at the instant the bottom of the ladder is 6 feet from the wall
Mathematics
1 answer:
Alex777 [14]3 years ago
5 0

Answer:

The area is changing at 15.75 square feet per second.

Step-by-step explanation:

The triangle between the wall, the ground, and the ladder has the following dimensions:

H: is the length of the ladder (hypotenuse) = 10 ft

B: is the distance between the wall and the ladder (base) = 6 ft

L: the length of the wall (height of the triangle) =?

dB/dt = is the variation of the base of the triangle = 9 ft/s        

First, we need to find the other side of the triangle:  

H^{2} = B^{2} + L^{2}

L = \sqrt{H^{2} - B^{2}} = \sqrt{(10)^{2} - B^{2}} = \sqrt{100 - B^{2}}

Now, the area (A) of the triangle is:            

A = \frac{BL}{2}  

Hence, the rate of change of the area is given by:

\frac{dA}{dt} = \frac{1}{2}[L*\frac{dB}{dt} + B\frac{dL}{dt}]      

\frac{dA}{dt} = \frac{1}{2}[\sqrt{100 - B^{2}}*\frac{dB}{dt} + B\frac{d(\sqrt{100 - B^{2}})}{dt}]        

\frac{dA}{dt} = \frac{1}{2}[\sqrt{100 - B^{2}}*\frac{dB}{dt} - \frac{B^{2}}{(\sqrt{100 - B^{2}})}*\frac{dB}{dt}]  

\frac{dA}{dt} = \frac{1}{2}[\sqrt{100 - 6^{2}}*9 - \frac{6^{2}}{\sqrt{100 - 6^{2}}}*9]      

\frac{dA}{dt} = 15.75 ft^{2}/s  

     

Therefore, the area is changing at 15.75 square feet per second.

I hope it helps you!                                    

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Kent has two similar cylindrical pipes, pipe a and pipe b. the radius of pipe a is 6 cm, and the radius of pipe b is 2 cm. what
White raven [17]

The ratio of the volume of pipe a to the volume of pipe b is 27:1  .

<h3>What is cylinder?</h3>

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder.

Given that,

Radius of pipe a is 6 cm, and the radius of pipe b is 2 cm.

We know that the volume of cylinder is :- \pi r^{2}h

where r is radius and h is height of the cylinder.

If two figures are similar then ratio of volume is equal to the cube of any dimension.

The ratio of the volume of Pipe a to the volume of Pipe b  :-

\frac{V(a)}{V(b)}=\frac{6^{3} }{2^{3} }

       = \frac{27}{1}

Hence, The ratio of the volume of pipe a to the volume of pipe b is 27:1 .

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7 0
1 year ago
Fred and Ben each ran laps around the track. Fred completed his run in 16 minutes. Ben completed his run in 3/4 of the amount of
m_a_m_a [10]

Answer:

Time required by Ben was 12 minutes.

Step-by-step explanation:

Given:

Time Required for Fred = 16 mins

Ben completed his run in 3/4 of the amount of time of Freds run.

We need to find the time required by Ben.

Now Given that;

Ben completed his run in 3/4 of the amount of time of Freds run.

It means that time required by Ben is equal to 3/4  times time required by Fred.

Time required by Ben = \frac{3}{4}\times  Freds \ run

Time required by Ben = \frac{3}{4}\times 16 = 12\ mins

Hence Time required by Ben was 12 minutes.

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vladimir2022 [97]

The equation that best describes the distance traveled is y = 190 + 65x.

Finding the values of the variables that lead to the stated equality being true is a step in the process of solving an equation with variables. Equation solutions are the amounts of the unknowns that meet the equality, whereas unknowns are the variables with which the equation must be solved.

Identity equations and conditional equations are the two types of equations. For every possible value of a variable, an identity holds true. The variables in a conditional equation can only have certain values.

An equation is made up of two expressions connected together by an equals symbol ("=").

The Distance traveled by car is 190 miles

The Speed of the car is 65 miles per hour.

Let x be the total number of hours traveling by car and y be the total distance remaining.

Then,

Total distance = 190 miles + 65 miles per hour × Number of hours traveled.

y = 190 + 65x

Consequently, the equation is y = 190 + 65 x.

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An online shopping club has 10,000 members when it charges $7 per month for membership. For each $1 monthly
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Answer:

The function that can be used in the online shopping club about its monthly revenue is:

R = -400x^{2} +7200x+70000

Step-by-step explanation:

First, we're gonna take into account the different values we have in the exercise:

  1. 10,000 members
  2. $7 per month for membership
  3. Loses of 400 members by each $1 monthly increase

How the variable x represents the price increase, we can do the formula below:

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In this formula, we represent in the first part that by each 1 in the variable x, the total of members will be reduced in 400, in the second part, we mention that at the same time, the membership fee will be increased in the same value of x. Now we must simplify this function:

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We operate the values:

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Solve we can:

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And organize:

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At the end, how R represents the monthly revenue received by the club, we use that variable for our formula:

  • R=-400x^{2} +7200x+70000

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