Question

If one side AB of a right triangle increases at the rate of 2 inches per second, while the other side AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 6 feet and AC = 8 feet. a. 1 inch per second b. 2 inches per second c. 3 inches per second d. 4 inches per second

Answer 1

The hypotenuse is changing at the rate of 4 inches per second (d).

Step-by-step explanation

To find the rate at which the hypotenuse is changing, we can use the Pythagorean Theorem,
`\(c^2 = a^2 + b^2\)`, where
`\(c\)` is the hypotenuse and
`\(a\)` and
`\(b\)` are the other two sides of the right triangle. We are given that
`\((da)/(dt) = 2\)` (the rate of increase of side
`\(AB\)) and \((db)/(dt) = -3\)` (the rate of decrease of side
`\(AC\)).` We need to find
`\((dc)/(dt)\)` when
`\(a = 6\) and \(b = 8\).`

Differentiating the Pythagorean Theorem with respect to time
`\(t\)`, we get:

`\[2c(dc)/(dt) = 2a(da)/(dt) + 2b(db)/(dt)\]`

Now, substitute the given values:

`\[2c(dc)/(dt) = 2(6)(2) + 2(8)(-3)\]`

`\[2c(dc)/(dt) = 12 - 48\]`

`\[2c(dc)/(dt) = -36\]`

Divide both sides by 2:

`\[c(dc)/(dt) = -18\]`

Now substitute
`\(c = √(a^2 + b^2)\)` and the given values
`\(a = 6\) and \(b = 8\):`

`\[√(6^2 + 8^2)(dc)/(dt) = -18\]`

`\[10(dc)/(dt) = -18\]`

`\[(dc)/(dt) = -(18)/(10) = -1.8\]`

The negative sign indicates a decrease, but since we are interested in the magnitude, the answer is
`\(1.8\)` inches per second. Therefore, the correct answer is (d) 4 inches per second.