The sides of an equilateral triangle are increasing at the rate of 5 centimeters per hour. At what rate is the area increasing when the side is 10 centimeters?
I have answered 4 of your 5 Calculus questions now.
Show me in this one that you have some kind of idea as to how to solve it.
50 cm2/hr
To find the rate at which the area is increasing, we can use the formula for the area of an equilateral triangle, which is given by:
Area = (√3/4) * side^2
where "side" represents the length of one side of the equilateral triangle.
Given that the sides of the equilateral triangle are increasing at a rate of 5 centimeters per hour, we can differentiate both sides of the area formula with respect to time (t):
d(Area)/dt = d(√3/4 * side^2)/dt
Now, let's substitute the given rate of change of the side (ds/dt = 5 cm/hr) and the value of the side when the area is desired (side = 10 cm) into the equation:
d(Area)/dt = d(√3/4 * (10 cm)^2)/dt
d(Area)/dt = d(√3/4 * 100 cm^2)/dt
d(Area)/dt = (√3/4) * 2 * 100 * d(side)/dt
Since we know d(side)/dt = 5 cm/hr, we can substitute this into the equation:
d(Area)/dt = (√3/4) * 2 * 100 * (5 cm/hr)
Simplifying further:
d(Area)/dt = (√3/4) * 2 * 100 * 5 cm/hr
Now, we can calculate the rate at which the area is increasing:
d(Area)/dt = (√3/4) * 2 * 100 * 5 cm/hr
d(Area)/dt = (√3/4) * 2 * 100 * 5 cm/hr
d(Area)/dt = (√3/4) * 1000 cm^2/hr
So, when the side of the equilateral triangle is 10 centimeters, the rate at which the area is increasing is (√3/4) * 1000 cm^2/hr.