The side of a square box is increasing at a rate of 4mm/s and its height is increasing at a rate of 2mm/s. How fast is the volume increasing when the side is 40mm and the height is 100mm.
Can someone just tell me what formula to use?
Don't always look for a formula.
Define your variables and "create" your own formula
let the square base have sides x by x
let the height be h
Volume = x^2 h ----- (from V = length x width x height)
using the product rule:
d(Volume)/dt = x^2 dh/dt + h(2x) dx/dt
we are given all those values
dV/dt = 40^2 (2) + 100(80)(4)
= ....
To find the rate at which the volume is increasing, you can use the formula for the volume of a rectangular box: V = lwh, where V is the volume, l is the length, w is the width, and h is the height.
In this case, the box is a square, so the length (l) and width (w) are equal to each other. Let's call the side length of the square box "s".
Now, we are given that the side length (s) is increasing at a rate of 4 mm/s, and the height (h) is increasing at a rate of 2 mm/s. We need to find how fast the volume (V) is changing when s = 40 mm and h = 100 mm.
To do this, we can differentiate the volume equation with respect to time (t) using the chain rule:
dV/dt = dV/ds * ds/dt + dV/dh * dh/dt
Here, dV/dt represents the rate at which the volume is changing, ds/dt represents the rate at which the side length is changing, and dh/dt represents the rate at which the height is changing.
Now, we need to find dV/ds and dV/dh. By differentiating the volume equation, we get:
dV/ds = wh, and dV/dh = lw
Since the box is a square, l = w = s. Therefore, dV/ds = sh and dV/dh = ls.
Now, let's substitute the given values and calculate the values of dV/ds and dV/dh:
When s = 40 mm and h = 100 mm,
dV/ds = sh = 40 * 100 = 4000 mm^2,
dV/dh = ls = 40 * 40 = 1600 mm^2.
Now, let's find the rate of change of volume dV/dt:
dV/dt = dV/ds * ds/dt + dV/dh * dh/dt
= 4000 mm^2 * 4 mm/s + 1600 mm^2 * 2 mm/s
= 16000 mm^3/s + 3200 mm^3/s
= 19200 mm^3/s.
Therefore, the volume is increasing at a rate of 19200 mm^3/s when the side length is 40 mm and the height is 100 mm.