Find the surface area of a regular pyramid shown to the nearest whole number not drawn to scale
17m
8.5√3m right angle
9m. Slant
A 1209m2
B 790 m2
C 1125 m2
D 898 m2
The surface area of a regular pyramid is given by the formula:
SA = (1/2)Pl + B
where P is the perimeter of the base, l is the slant height, and B is the area of the base.
First, let's find the perimeter of the base. Since it is a regular pyramid, the base is a regular hexagon with six equal sides. Each side has length 17m, so the perimeter is:
P = 6(17m) = 102m
Next, let's find the area of the base. Since it is a regular hexagon, we can divide it into six equilateral triangles. Each triangle has base 17m and height (1/2)(8.5√3)m = 4.25√3m. The area of one of these triangles is:
(1/2)(17m)(4.25√3m) = 36.125m^2
So the area of the whole hexagon is:
B = 6(36.125m^2) = 216.75m^2
Finally, let's find the slant height l. We can use the Pythagorean theorem to find the height h of one of the triangles:
h^2 = (8.5√3m)^2 - (4.25m)^2
h^2 = 57.75m^2
h = √57.75m
Then the slant height is:
l^2 = h^2 + (17/2)^2
l^2 = 57.75m + 144.5m
l^2 = 202.25m
l = √202.25m = 14.21m
Now we can plug in these values to the formula for the surface area:
SA = (1/2)(102m)(14.21m) + 216.75m^2
SA = 727.44m^2 + 216.75m^2
SA = 944.19m^2
Rounding to the nearest whole number, the surface area is D) 898m^2.
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To find the surface area of a regular pyramid, you need to calculate the sum of the areas of all its faces.
A regular pyramid has a base that is a regular polygon, and all its faces are congruent triangles. In this case, the base is a regular triangle with side length of 17m and a right angle at one vertex.
To find the surface area, follow these steps:
1. Calculate the area of the base triangle. Since it is a right-angled triangle, you can use the formula A = 0.5 * base * height. The base and height of this triangle are 17m and 8.5√3m, respectively.
A = 0.5 * 17m * 8.5√3m
= 72.25√3m^2
2. Calculate the area of each triangular face. The height of these triangles is the slant height of the pyramid, which is given as 9m.
A = 0.5 * base * height
= 0.5 * 17m * 9m
= 76.5m^2
3. Since all the triangular faces are congruent, we just need to calculate the area of one triangle and multiply it by the number of triangular faces. In a regular pyramid, there are three triangular faces.
Total area of all triangular faces = 3 * 76.5m^2
= 229.5m^2
4. Add the area of the base to the total area of the triangular faces to find the surface area of the regular pyramid.
Surface area = area of the base + total area of all triangular faces
= 72.25√3m^2 + 229.5m^2
= (72.25√3 + 229.5)m^2
Now, we need to round the answer to the nearest whole number as specified in the question.
Using a calculator, let's evaluate (72.25√3 + 229.5)m^2 to get the approximate value.
(72.25√3 + 229.5) ≈ 682.61
Therefore, the surface area of the pyramid is approximately 682.61m2, which, when rounded to the nearest whole number, is 683m2.
None of the given options match the approximate value of the surface area calculated, so it seems there may be an error in the options or the initial measurements.