Questions: The formula for finding the volume of a square pyramid is V = 1/3 s^2 h. The length of the side of a model pyramid is (x+2) cm. The height of the pyramid is (x-1) cm. If the volume of the pyramid is 12 cm^3, determine the length of the side and the height of the pyramid. [4marks]

Solution Steps

To solve for the length of the side and the height of the pyramid, we need to set up an equation using the given volume formula and solve for \( x \). Once we find \( x \), we can determine the side length and height.

1. Substitute the given expressions for the side length and height into the volume formula.
2. Set the equation equal to the given volume.
3. Solve the resulting equation for \( x \).
4. Use the value of \( x \) to find the side length and height.

We start with the volume formula for a square pyramid given by

\[ V = \frac{1}{3} s^{2} h \]

where \( s \) is the side length and \( h \) is the height. Substituting the expressions for the side length \( s = x + 2 \) and height \( h = x - 1 \), we have:

\[ V = \frac{1}{3} (x + 2)^{2} (x - 1) \]

Setting this equal to the given volume of \( 12 \, \text{cm}^{3} \):

\[ \frac{1}{3} (x + 2)^{2} (x - 1) = 12 \]

Multiplying both sides by \( 3 \) to eliminate the fraction gives:

\[ (x + 2)^{2} (x - 1) = 36 \]

This equation can be solved for \( x \). The solutions obtained are:

\[ x \approx 2.6587, \quad x \approx -2.8294 - 2.6532i, \quad x \approx -2.8294 + 2.6532i \]

Since \( x \) must be a real number for the dimensions of the pyramid, we only consider the real solution:

\[ x \approx 2.6587 \]

Using the valid solution for \( x \):

• Side length \( s \):

\[ s = x + 2 \approx 2.6587 + 2 \approx 4.6587 \, \text{cm} \]

• Height \( h \):

\[ h = x - 1 \approx 2.6587 - 1 \approx 1.6587 \, \text{cm} \]

Final Answer