Education.com
Try
Brainzy
Try
Plus

# Differentiate Both Sides of the Equation Study Guide (page 2)

based on 1 rating
By
Updated on Oct 1, 2011

#### Solution 7

If we represent the length of the base by b, the height of the triangle as h, and the area of the triangle as A, then the formula that relates them all is A = . The base is increasing a and the height is changing at .The –1 implies that 1 foot is subtracted from the height every minute, that is, the height is decreasing. We are trying to find , which is the rate of change in area. When we differentiate our formula A = with respect to t, we get:

When we plug in all of our information, including the h = 5 feet and b = 20 feet, we get:

= 20 – 10 = 10

Thus, at the exact instant when the height is 5 feet and the base is 20, the area of the triangle is increasing at a rate of 10 square feet every minute.

#### Example 8

A 20 foot ladder slides down a wall at the rate of 2 feet per minute (see Figure 12.1). How fast is it sliding along the ground when the ladder is 16 feet up the wall?

#### Solution 8

Here, because the ladder is sliding down the wall at 2 feet per minute. We want to know , the rate at which the bottom of the ladder is moving away from the wall. The equation to use is the Pythagorean theorem.

x2 + y2 = 202

(x2 + y2) = (202)

2x · + 2y · = 0

If we plug in y = 16 and = –2, we get:

2x · + 2(16) · (–2) = 0

We still need to know what x is at the particular instant that y = 16, and for this, we go back to the Pythagorean theorem.

x2 + (16)2 = (20)2

x2 = 144, so x = ±12

Using x = 12 (a negative length here makes no sense), we get:

2 · (12) · + 2 · (16)(–2) = 0

=

At the moment that y = 16, the ladder is sliding along the ground at feet per minute.

In the previous example, it was okay to say that the hypotenuse was 20 because the length of the ladder didn't change. However, if we replace y with 16 in the equation before differentiating, we would have implied that the height was fixed at 16 feet. Because the height does change, it needs to be written as a variable, y. In general, anything that varies needs to be represented with a variable. Only after the derivative has been taken can the information for the given instant, like y = 16, be substituted.

### Changing Values Hint

It is important to use variables for all of the values that are changing. Only after differentiation can they be replaced by numbers.

Find practice problems and solutions for these concepts at Differentiate Both Sides of the Equation Practice Questions

150 Characters allowed

### Related Questions

#### Q:

See More Questions

### Today on Education.com

Top Worksheet Slideshows