at a certain instant the base of a triangle is 5 inches and is increasing at the rate of 1 inch per minute. At the same instant, the height is 10 inches and is decreasing at the rate of 2.5 inches per minute. Is the area of the triangle increasing or decreasing? justify you answer.

a = bh/2

da/dt = h/2 db/dt + b/2 dh/dt
= 10/2 (1) + 5/2 (-2.5)
= -1.25 in^2/min

Ok. Just do it the brute-force way.

Initially, the area is 1/2 x 5 x 10 = 25
After one minute, the area is 1/2 x 6 x 7.5 = 22.5
After two minutes, the area is 1/2 x 7 x 5 = 17.5

You can see that the area is decreasing.

In terms of derivatives, the rate of change is exactly what a derivative is.

You can say +1 d/dx for the base and -2.5 d/dx for the height.

Overall that's a decrease in 1 - 2.5 = -1.5 inches every minute.

If you look at the equation for a triangle 1/2 x b x h, the overall value will be less because you're losing -1.5 inches overall.

Well, let's clown around with some math, shall we?

To determine if the area of the triangle is increasing or decreasing, we need to look at how the area changes with respect to time. The formula for the area of a triangle is (*drumroll, please*) A = (1/2) * base * height.

Since the base is increasing at a rate of 1 inch per minute, we can represent this as dB/dt = 1. Similarly, since the height is decreasing at a rate of 2.5 inches per minute, we have dH/dt = -2.5.

Now, let's differentiate our area formula with respect to time using the product rule. dA/dt = (1/2) * (base * dH/dt + height * dB/dt).

Substituting in the given values, we get: dA/dt = (1/2) * (5 * (-2.5) + 10 * 1). Simplifying this expression, we have dA/dt = -6.25 + 5 = -1.25.

Since the derivative of the area is negative (-1.25), this means that the area of the triangle is decreasing at a rate of 1.25 square inches per minute.

So, to sum it all up: the area of the triangle is decreasing!

To determine whether the area of the triangle is increasing or decreasing, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Given that the base of the triangle is increasing at a rate of 1 inch per minute and the height is decreasing at a rate of 2.5 inches per minute, we can calculate the rates of change for each component of the area formula.

The rate of change of the area can be found by taking the derivative with respect to time. Let's calculate the rates of change.

d(base)/dt = 1 inch per minute (rate of change of the base)
d(height)/dt = -2.5 inches per minute (rate of change of the height)

Applying these rates of change to the formula for the area of a triangle, we have:

d(Area)/dt = (1/2) * (base * d(height)/dt + height * d(base)/dt)

Substituting the given values:

d(Area)/dt = (1/2) * (5 * -2.5 + 10 * 1)

Simplifying:

d(Area)/dt = (1/2) * (-12.5 + 10)

d(Area)/dt = (1/2) * (-2.5)

d(Area)/dt = -1.25

The rate of change of the area is -1.25 square inches per minute.

Since the rate of change of the area is negative, the area of the triangle is decreasing.

To determine whether the area of the triangle is increasing or decreasing, we need to find the rate of change of its area.

The formula for the area of a triangle is given by: Area = (1/2) * base * height

Given that the base is increasing at a rate of 1 inch per minute, we can represent it as db/dt = 1 in/min. Similarly, the height is decreasing at a rate of 2.5 inches per minute, which can be represented as dh/dt = -2.5 in/min.

To find the rate of change of the area, we differentiate the area formula with respect to time using the product rule:

dA/dt = (1/2) * (db/dt) * height + (1/2) * base * (dh/dt)

Substituting the given values:

dA/dt = (1/2) * (1 in/min) * 10 in + (1/2) * 5 in * (-2.5 in/min)

Simplifying:

dA/dt = 5 in/min - 6.25 in/min
dA/dt = -1.25 in/min

The negative sign indicates that the rate of change of the area is negative, which means the area of the triangle is decreasing.

Justification: Since the base of the triangle is increasing at a slower rate than the height is decreasing, the decrease in height has a greater effect on the area of the triangle, leading to a net decrease in the area over time.