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)
Transcript text: Find the surface area of a rectangular pyramid with these measurements: $l=$ $15 \mathrm{~m}, w=12 \mathrm{~m}$, and $h=10 \mathrm{~m}$. Express your answer as a decimal rounded to the nearest hundredth. (1 point)
Solution
Solution Steps
Step 1: Calculate the Area of the Base
The area of the rectangular base of the pyramid is given by the formula:
Base Area=l×w=15m×12m=180m2
Step 2: Calculate the Slant Heights
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 \( w \):
\[
\text{Slant Height}_w = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{7.5^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25}
\]
Step 3: Calculate the Area of the Triangular Faces
The area of each triangular face can be calculated using the formula:
Triangular Areal=21×l×Slant HeightlTriangular Areaw=21×w×Slant Heightw
Substituting the values:
Triangular Areal=21×15×136Triangular Areaw=21×12×156.25
Step 4: Calculate the Total Surface Area
The total surface area of the pyramid is the sum of the base area and the areas of the four triangular faces:
Surface Area=Base Area+2×Triangular Areal+2×Triangular Areaw
Substituting the calculated areas:
Surface Area=180+2×(21×15×136)+2×(21×12×156.25)
Step 5: Round the Result
Finally, round the total surface area to the nearest hundredth to express the answer in the required format.