How Fast is the Triangle's Area Changing When the Board is 9 Feet from the Wall?

In summary: I'll leave it to you to fill in the gap.In summary, the problem involves finding the rate of change of the area of a right triangle formed by a 15 foot board leaning against a wall, with its bottom sliding away from the wall at a rate of 3 feet per second. Using the relationship A = 1/2x√(225-x^2), where x represents the horizontal distance between the bottom of the ladder and the wall, the rate of change of A can be found by taking the derivative and evaluating it at the given value of x. The final answer will be in square feet per second.
  • #1
buffgilville
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What is the y prime of:
A 15 foot board rests against a vertical wall. If the bottom of the board slides away from the wall at the rate of 3 feet per second, how fast is the area of the triangle formed by the board, the wall and the ground changing at the instant the bottom of the board is 9 feet from the wall?

I got this far:
y= square root of (225 - x^2)

I'm stuck after this step.

Can the answer for this be [(-9 yprime /2) + 18] square feet per second.
 
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  • #2
Hint x in terms of t. The displacement x away from the wall is equal to what in terms of t? It will be easy to then sub in this for x.

Area of a triangle is 1/2 * base * height
Base is x height is sqrt(225-x^2) right?

so A = 1/2 * x * sqrt(225-x^2)

However you need to plug in t to this I just didn't want to give away that much work.

Differentiate with respect to t in the new equation you find using the product rule (gets a little messy). Since 3t = 9, t = 3 goes into this new derivative and bam you got your answer.
 
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  • #3
I'm not sure what you mean by the question "what is the y prime of...?". If anything, the question seems to be asking for A'(t), where A is the area of the right triangle formed by the ladder, the wall, and the ground.

Draw a diagram, label the horizontal distance between the bottom of the ladder and wall x, and the corresponding vertical distance y

Given:

[itex] \frac{dx}{dt} [/itex] = 3 ft/s

This is a related rates problem. A is a function of x, so given the rate at which x changes with time, you should be able to find the rate at which A changes with time using the functional relationship:

[tex] A = \frac{1}{2}xy = \frac{1}{2}x\sqrt{225 - x^2} [/tex]

You identified this relationship correctly! :smile:

Now, please relate the rates:

[tex] \frac{dA}{dt} = \frac{dA}{dx}\frac{dx}{dt} [/tex]

[tex] = \frac{1}{2}\frac{d}{dx}(x\sqrt{225 - x^2})(3 \text{ft/s}) [/tex]

Once you solve that, evaluate it at the value specified. I think that's right. It's kinda late here, so if any of this is bogus, please let me know.

EDIT: right, vsage, I forgot that you need x in terms of t.
 
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FAQ: How Fast is the Triangle's Area Changing When the Board is 9 Feet from the Wall?

What is the purpose of finding y prime (derivative)?

The purpose of finding y prime, also known as taking the derivative, is to determine the rate of change of a function. It allows us to analyze and understand the behavior of a function at any point by calculating its slope or instantaneous rate of change.

How do you find the derivative of a function?

The derivative of a function can be found using the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules involve taking the limit as the change in the input variable approaches zero. Alternatively, you can also use a graphing calculator or software to find the derivative.

What does the derivative tell us about a function?

The derivative tells us about the slope of a function at any given point. It also represents the rate of change of the function, which can be interpreted as the speed or direction of the function's growth or decline. Furthermore, the derivative can help us identify critical points, such as maximum and minimum values, and concavity of the function.

Can the derivative ever be undefined?

Yes, the derivative can be undefined at certain points. This occurs when the function is not continuous or has a sharp turn or corner at that point. It can also happen when the function has a vertical tangent line, which indicates a sharp change in the slope of the function at that point.

Why is the derivative important in real-world applications?

The derivative has many real-world applications, including physics, engineering, economics, and statistics. It allows us to model and analyze the behavior of various systems, such as the motion of objects, the growth of populations, and the changes in market trends. It also helps us optimize functions and make predictions based on current data.

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