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A pyramid with a square base has a fixed volume. Its height varies inversely as the square of the length of side of its base. The length of the side of its base is 4 cm when its height is 5 cm. Calculate the length of side of the base, in cm, of the pyramid with a height of 3.2 cm.​

User Lmm
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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

User Stepan Suvorov
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