The altitude of a triangle is increasing at a rate of 2500 centimeters/minute while the area of the triangle is increasing at a rate of 4000 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 11500 centimeters and the area is 89000 square centimeters?Note: The "altitude" is the "height" of the triangle in the formula "Area=(1/2)baseheight". Draw yourself a general "representative" triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don't need to know how big nor what shape the triangle really is.

Question

The altitude of a triangle is increasing at a rate of 2500 centimeters/minute while the area of the triangle is increasing at a rate of 4000 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 11500 centimeters and the area is 89000 square centimeters?Note: The "altitude" is the "height" of the triangle in the formula "Area=(1/2)baseheight". Draw yourself a general "representative" triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don't need to know how big nor what shape the triangle really is.

Answer 1

To find the rate at which the base of the triangle is changing, we need to use the chain rule of calculus. Let's solve this step by step.

Step 1: Identify the given information.
- The altitude (height) of the triangle is increasing at a rate of 2500 centimeters/minute.
- The area of the triangle is increasing at a rate of 4000 square centimeters/minute.
- The altitude is 11500 centimeters.
- The area is 89000 square centimeters.

Step 2: Identify the variables.
- Let's assume the base of the triangle as "b" and the altitude as "h." These will be the variables we will work with.

Step 3: Write down the given information.
- We have h = 11500 cm and the area, A = 89000 cm².

Step 4: Write down the formula for the area of the triangle.
- The formula for the area of a triangle is: A = (1/2) * b * h.

Step 5: Find the formula for the rate of change of the area with respect to time.
- Differentiate the area formula with respect to time (t) using the chain rule of calculus:
dA/dt = (1/2) * b * dh/dt + (1/2) * h * db/dt.

Step 6: Substitute the given values and the desired rate of change.
- We know that dA/dt = 4000 cm²/min and dh/dt = 2500 cm/min.
- Substituting these values, along with h = 11500 cm and A = 89000 cm², into the formula:
4000 = (1/2) * b * 2500 + (1/2) * 11500 * db/dt.

Step 7: Solve for db/dt.
- Simplifying the equation:
4000 = 1250b + 5750 * db/dt.
- Rearranging the terms to isolate db/dt:
5750 * db/dt = 4000 - 1250b.
db/dt = (4000 - 1250b) / 5750.

So, the rate at which the base of the triangle is changing when the altitude is 11500 cm and the area is 89000 cm² is given by the equation db/dt = (4000 - 1250b) / 5750.