Answer:
The length of the side of the base of the pyramid with a height of 3.2 cm is 5 cm.
Explanation:
Let V be the fixed volume of the pyramid, and let h and s be the height and length of a side of its base, respectively. Then we have:
V = (1/3) * s^2 * h ... (1)
Also, we have:
h ∝ 1/s^2 ... (2)
We are given that s = 4 cm when h = 5 cm. Using this information, we can find the constant of proportionality k in equation (2) as follows:
5 ∝ 1/4^2
5 ∝ 1/16
k = 16 * 5 = 80
Therefore, we have:
h = k/s^2 ... (3)
Now we can use equations (1) and (3) to find the length of the side of the base when the height is 3.2 cm:
V = (1/3) * s^2 * h
V = (1/3) * s^2 * (k/s^2)
V = k/3 * s^2
We know that V is fixed, so we can set the right-hand side of this equation equal to V and solve for s:
V = k/3 * s^2
s^2 = 3V/k
s = sqrt(3V/k)
Plugging in the given values, we get:
s = sqrt(3V/80)
s = sqrt(3*V)/sqrt(80)
s = sqrt(3)/4 * sqrt(V)
Now we can use the given height of 3.2 cm to find the value of V, and then substitute it into the equation for s:
3.2 = k/s^2
s^2 = k/3.2
s = sqrt(k/3.2)
Plugging in the value of k we found earlier, we get:
s = sqrt(80/3.2)
s = sqrt(25)
s = 5