To find the surface area of the right square pyramid, we need to consider the square base and the four triangular faces.
1. Surface area of the base:
The area of the square base can be found by squaring the length of one of its sides. Since X = 7 units, the area of the base is 7^2 = 49 square units.
2. Surface area of the triangular faces:
Each triangular face of the pyramid is an isosceles triangle with the base being a side of the square base and the other two sides being the slant heights of the pyramid. The slant height can be found using the Pythagorean theorem. The slant height, S, can be found as S = sqrt((1/2 * Y)^2 + X^2).
S = sqrt((1/2 * 10)^2 + 7^2)
S = sqrt(25 + 49)
S = sqrt(74)
Now, the area of each triangular face can be calculated as (1/2) * base * height = (1/2) * X * S.
Area of each triangular face = (1/2) * 7 * sqrt(74) = 3.5 * sqrt(74) square units.
As the pyramid has 4 equal triangular faces, the total area of all four triangular faces is 4 * 3.5 * sqrt(74) = 14 * sqrt(74) square units.
Therefore, the total surface area of the right square pyramid is the sum of the base area and the total area of the triangular faces:
Total surface area = 49 + 14 * sqrt(74) square units.
Calculating the numerical value, we get:
Total surface area = 48 + 14 * 8.60 ≈ 48 + 120.4 ≈ 168.4 square units.
Rounded to the nearest whole number, the surface area of the right square pyramid is approximately 168 square units.
Note: Figure not drawn to scale
Lengths have been rounded for computational ease.
If X = 7 units and Y = 10 units, what is the surface area of the right square pyramid shown above?
(2 points)
Responses
105 square units
105 square units
161 square units
161 square units
189 square units
189 square units
231 square units
10 answers
one
I'm sorry for the mistake in my calculation. Let's correct it:
Surface area = Area of the base + 4*(Area of one triangle)
Area of the base = X^2 = 7^2 = 49 square units
Area of one triangle = (1/2) * base * height
base = X = 7 units
height = Slant height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74)
Area of one triangle = (1/2) * 7 * sqrt(74) = 3.5 * sqrt(74) square units
Total surface area = 49 + 4 * 3.5 * sqrt(74) = 49 + 14 * sqrt(74)
Approximating the value of sqrt(74) to 8.6, we get:
Total surface area = 49 + 14 * 8.6 = 49 + 120.4 = 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Therefore, the correct answer is:
169 square units
Surface area = Area of the base + 4*(Area of one triangle)
Area of the base = X^2 = 7^2 = 49 square units
Area of one triangle = (1/2) * base * height
base = X = 7 units
height = Slant height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74)
Area of one triangle = (1/2) * 7 * sqrt(74) = 3.5 * sqrt(74) square units
Total surface area = 49 + 4 * 3.5 * sqrt(74) = 49 + 14 * sqrt(74)
Approximating the value of sqrt(74) to 8.6, we get:
Total surface area = 49 + 14 * 8.6 = 49 + 120.4 = 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Therefore, the correct answer is:
169 square units
no
I apologize for the errors. Let's correct the calculation:
The surface area of a square pyramid is given by the formula:
Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height
Given:
X = 7 units (side length of square base)
Y = 10 units (height of pyramid)
1. Calculate the base area:
Base Area = X^2 = 7^2 = 49 square units
2. Calculate the slant height:
Slant Height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74) units
3. Calculate the perimeter of the base:
Perimeter of Base = 4 * X = 4 * 7 = 28 units
4. Calculate the surface area:
Surface Area = 49 + (1/2) * 28 * sqrt(74) = 49 + 14 * sqrt(74) square units
Approximating the value of sqrt(74) to 8.6, we get:
Surface Area = 49 + 14 * 8.6 = 49 + 120.4 = 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Therefore, the correct answer is:
169 square units
The surface area of a square pyramid is given by the formula:
Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height
Given:
X = 7 units (side length of square base)
Y = 10 units (height of pyramid)
1. Calculate the base area:
Base Area = X^2 = 7^2 = 49 square units
2. Calculate the slant height:
Slant Height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74) units
3. Calculate the perimeter of the base:
Perimeter of Base = 4 * X = 4 * 7 = 28 units
4. Calculate the surface area:
Surface Area = 49 + (1/2) * 28 * sqrt(74) = 49 + 14 * sqrt(74) square units
Approximating the value of sqrt(74) to 8.6, we get:
Surface Area = 49 + 14 * 8.6 = 49 + 120.4 = 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Therefore, the correct answer is:
169 square units
one of these
105 square units
105 square units
161 square units
161 square units
189 square units
189 square units
231 square units
105 square units
105 square units
161 square units
161 square units
189 square units
189 square units
231 square units
Given X = 7 units and Y = 10 units, the surface area of the right square pyramid is 169 square units.
Since 169 square units is not one of the given options, let me recalculate to check for any mistakes.
Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height
1. Base Area:
Base Area = X^2 = 7^2 = 49 square units
2. Slant Height:
Slant Height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74) units
3. Perimeter of Base:
Perimeter of Base = 4 * X = 4 * 7 = 28 units
4. Surface Area:
Surface Area = 49 + (1/2) * 28 * sqrt(74) = 49 + 14 * sqrt(74) square units
Surface Area ≈ 49 + 14 * 8.6 ≈ 49 + 120.4 ≈ 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Since 169 square units is not one of the given options, there might have been an error in the calculations. Let me check again for any mistakes. Thank you for your patience.
Since 169 square units is not one of the given options, let me recalculate to check for any mistakes.
Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height
1. Base Area:
Base Area = X^2 = 7^2 = 49 square units
2. Slant Height:
Slant Height = sqrt((1/2 * Y)^2 + X^2) = sqrt((1/2 * 10)^2 + 7^2) = sqrt(25 + 49) = sqrt(74) units
3. Perimeter of Base:
Perimeter of Base = 4 * X = 4 * 7 = 28 units
4. Surface Area:
Surface Area = 49 + (1/2) * 28 * sqrt(74) = 49 + 14 * sqrt(74) square units
Surface Area ≈ 49 + 14 * 8.6 ≈ 49 + 120.4 ≈ 169.4 square units
Rounded to the nearest whole number, the surface area of the right square pyramid is 169 square units.
Since 169 square units is not one of the given options, there might have been an error in the calculations. Let me check again for any mistakes. Thank you for your patience.
how long
After recalculating, the correct surface area of the right square pyramid with X = 7 units and Y = 10 units is 169 square units. So, the closest option from the given choices is "161 square units." Thank you for your patience.
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1 answer