Questions: Find the surface area of a rectangular pyramid with these measurements: l= 15 m, w=12 m, and h=10 m. Express your answer as a decimal rounded to the nearest hundredth. (1 point)

Solution Steps

The area of the rectangular base of the pyramid is given by the formula: \[ \text{Base Area} = l \times w = 15 \, \text{m} \times 12 \, \text{m} = 180 \, \text{m}^2 \]

To find the slant heights of the triangular faces, we use the Pythagorean theorem. For the triangular face with base \( l \): \[ \text{Slant Height}_l = \sqrt{\left(\frac{w}{2}\right)^2 + h^2 = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \] For the triangular face with base \( w \): \[ \text{Slant Height}_w = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{7.5^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25} \]

The area of each triangular face can be calculated using the formula: \[ \text{Triangular Area}_l = \frac{1}{2} \times l \times \text{Slant Height}_l \] \[ \text{Triangular Area}_w = \frac{1}{2} \times w \times \text{Slant Height}_w \] Substituting the values: \[ \text{Triangular Area}_l = \frac{1}{2} \times 15 \times \sqrt{136} \] \[ \text{Triangular Area}_w = \frac{1}{2} \times 12 \times \sqrt{156.25} \]

The total surface area of the pyramid is the sum of the base area and the areas of the four triangular faces: \[ \text{Surface Area} = \text{Base Area} + 2 \times \text{Triangular Area}_l + 2 \times \text{Triangular Area}_w \] Substituting the calculated areas: \[ \text{Surface Area} = 180 + 2 \times \left(\frac{1}{2} \times 15 \times \sqrt{136}\right) + 2 \times \left(\frac{1}{2} \times 12 \times \sqrt{156.25}\right) \]

Finally, round the total surface area to the nearest hundredth to express the answer in the required format.

Final Answer