Math / Calculus

QuestionA right triangle has legs of 33 inches and 44 inches whose sides are changing. The short leg is decreasing by $3 \mathrm{in} / \mathrm{sec}$ and the long leg is growing at $8 \mathrm{in} / \mathrm{sec}$ . What is the rate of change of the hypotenuse?

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The legs of the triangle are initially 33 inches and 44 inches.
3. The short leg is decreasing at a rate of $3 \, \text{in/sec}$ .
4. The long leg is increasing at a rate of $8 \, \text{in/sec}$ .
5. We need to find the rate of change of the hypotenuse.

STEP 2

1. Use the Pythagorean theorem to express the hypotenuse in terms of the legs.
2. Differentiate the Pythagorean theorem with respect to time to find the rate of change of the hypotenuse.
3. Substitute the given rates of change of the legs into the differentiated equation to solve for the rate of change of the hypotenuse.

STEP 3

The Pythagorean theorem for a right triangle states that $c^2 = a^2 + b^2$ , where $c$ is the hypotenuse, and $a$ and $b$ are the legs.
Initially, $a = 33$ inches and $b = 44$ inches.

STEP 4

Differentiate the equation $c^2 = a^2 + b^2$ with respect to time $t$ .
Using implicit differentiation, we get: $2c \frac{dc}{dt} = 2a \frac{da}{dt} + 2b \frac{db}{dt}$
Simplify to: $c \frac{dc}{dt} = a \frac{da}{dt} + b \frac{db}{dt}$

STEP 5

Substitute the given rates: $\frac{da}{dt} = -3 \, \text{in/sec}$ and $\frac{db}{dt} = 8 \, \text{in/sec}$ .
Calculate the initial hypotenuse $c$ using $c = \sqrt{33^2 + 44^2}$ .
$c = \sqrt{1089 + 1936} = \sqrt{3025} = 55 \, \text{inches}$
Substitute into the differentiated equation: $55 \frac{dc}{dt} = 33(-3) + 44(8)$
Calculate: $55 \frac{dc}{dt} = -99 + 352$
$55 \frac{dc}{dt} = 253$
Solve for $\frac{dc}{dt}$ : $\frac{dc}{dt} = \frac{253}{55}$
$\frac{dc}{dt} \approx 4.6 \, \text{in/sec}$
The rate of change of the hypotenuse is approximately:
$\boxed{4.6 \, \text{in/sec}}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms