Final answer:
To find the rate at which the height is changing in an equilateral triangle, we can use the chain rule. The rate at which the height is changing is (10/(sqrt(3) * s)) times the rate at which the area is changing.
Step-by-step explanation:
To find the rate at which the height is changing, we can use the fact that an equilateral triangle has all sides and angles equal. Let's denote the side length of the equilateral triangle as 's' and the height as 'h'. The formula for the area of an equilateral triangle is A = (sqrt(3)/4) * s^2, where 'A' is the area and 's' is the side length.
We are given that the area is increasing at a rate of 5 m^2/hr, so dA/dt = 5 m^2/hr. We want to find dh/dt, the rate at which the height is changing.
Using the chain rule, we can express dA/dt in terms of dh/dt:
dA/dt = (dA/ds) * (ds/dt) = 5 m^2/hr
Now, differentiate the area formula with respect to 's' to find dA/ds:
dA/ds = (sqrt(3)/4) * 2s
Substituting this into the expression for dA/dt, we get:
(sqrt(3)/4) * 2s * (ds/dt) = 5 m^2/hr
Simplifying, we have:
(sqrt(3)/2) * s * (ds/dt) = 5 m^2/hr
Now, we can solve for dh/dt:
dh/dt = (ds/dt) * (2/(sqrt(3) * s)) * 5 = (ds/dt) * (10/(sqrt(3) * s))
Therefore, the rate at which the height is changing is (10/(sqrt(3) * s)) times the rate at which the area is changing.