The altitude of a triangle is increasing at a rate of 1.500 centimeters/minute while the area of the triangle is increasing at a rate of 2.000 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 8.000 centimeters and the area is 98.000 square centimeters?

The base of the triangle is shrinking at a rate of centimeters per minute.

Step-by-step explanation:

The formula of the area of a triangle is given by the following expression:

Where:

- Area of the triangle, measured in square centimeters.

- Base of the triangle, measured in centimeters.

- Height of the triangle, measured in centimeters.

The base of the triangle is:

If and , the base of the triangle is:

The rate of change of the area of the triangle in time, measured in minutes, is obtained after differentiating by rule of chain and using deriving rules:

The rate of change of the base of the triangle is now cleared:

Given that , , and , the rate of change of the base of the triangle is:

The base of the triangle is shrinking at a rate of centimeters per minute.

26

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240