Question

The hypotenuse of a right triangle is 25 feet long The height of the right triangle is shrinking at a rate of 1 ft/week. What is the rate of change of the base of the triangle when the height is 7 feet?

Answer 1

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Answer:

  increasing 7/24 ft per week

Step-by-step explanation:

An expression for the length of the base can be written from the Pythagorean theorem relation between the sides of the triangle.

If the height is 7-x, where x is the number of weeks after the height is 7 feet, then the base length is ...

  b = √(25^2 -(7-x)^2) = √(625 -(49 -14x +x^2)) = √(-x^2 +14x +576)

Then the rate of change of b with respect to x is ...

  b' = (-2x +14)/(2√(-x^2 +14x +576))

When x=0, the rate of change of the base is ...

  b' = 14/(2√576) = 7/24

The base is increasing 7/24 ft per week when the height is 7 ft.