Use the image to answer the question.
An illustration shows a triangular prism with length 7 feet, width 4 feet, and perpendicular height of the triangular face 3 feet.
What is the total surface area of the two triangular bases of this tent?
The total surface area of the two triangular bases is 21 square feet.
To find this, we need to use the formula for the area of a triangle:
Area = 1/2 x base x height
The base of each triangular face is 4 feet (the width of the prism), and the height is 3 feet (the perpendicular height of the triangular face).
So the area of each triangular face is:
1/2 x 4 feet x 3 feet = 6 square feet
And since there are two triangular faces, we can add these areas together:
6 square feet + 6 square feet = 12 square feet
Therefore, the total surface area of the two triangular bases is 12 square feet.
Use the image to answer the question.
An illustration shows a triangular prism and a triangular pyramid. The edges that are not visible are marked as dashed lines. The triangular prism has its triangular face as the base. The area of the triangular face is labeled as upper B equals 10 inches squared. The length is 7 inches. The triangular pyramid has the triangular face as its base with the area labeled upper B equals 10 inches squared. The perpendicular height of the pyramid is 7 inches.
How does the volume of the prism compare to the volume of the pyramid?
(1 point)
Responses
The volume of the pyramid is three times as large as the volume of the prism.
The volume of the pyramid is three times as large as the volume of the prism.
The volume of the prism is the same as the volume of the pyramid.
The volume of the prism is the same as the volume of the pyramid.
The volume of the prism is three times as large as the volume of the pyramid.
The volume of the prism is three times as large as the volume of the pyramid.
The volume of the prism is 13 the size of the pyramid.
The volume of the prism is the same as the volume of the pyramid.
To find the volume of the prism, we need to multiply the area of the triangular base (10 square inches) by the length of the prism (7 inches):
Volume of prism = 10 in² x 7 in = 70 cubic inches
To find the volume of the pyramid, we need to use the formula:
Volume of pyramid = 1/3 x area of base x height
The area of the triangular base is also 10 square inches, and the height is given as 7 inches:
Volume of pyramid = 1/3 x 10 in² x 7 in = 23.33 cubic inches
Therefore, the volume of the prism (70 cubic inches) is equal to the volume of the pyramid (23.33 cubic inches), so the answer is that the volumes are the same, option (2).
Use the image to answer the question.
An illustration shows a triangle with sides measuring 21, 17, and 10. A perpendicular line, from the side measuring 21 to the opposite angle, measures 8. A right angle symbol is shown to the left of the perpendicular line.
A prism and a pyramid both have this triangular base. If both shapes have the same height and the volume of the prism is 1,092 cubic units, what is the volume of the pyramid?
(1 point)
Responses
1,092 cubic units
1,092 cubic units
3,276 cubic units
3,276 cubic units
728 cubic units
728 cubic units
364 cubic units
364 cubic units
To find the volume of the pyramid, we need to know the formula for the volume of a pyramid, which is:
Volume of pyramid = 1/3 x area of base x height
We know that the base of the pyramid is the same as the triangular base given in the problem, with sides measuring 21, 17, and 10. We also know that the height of the pyramid is the same as the height of the prism.
To find the area of the base, we can use Heron's formula, which gives the area of a triangle when we know the lengths of its sides. Heron's formula is:
Area = sqrt[s(s - a)(s - b)(s - c)]
where a, b, and c are the sides of the triangle, and s is the semiperimeter (half the perimeter):
s = (a + b + c)/2
In this case, we have:
a = 21, b = 17, c = 10
s = (21 + 17 + 10)/2 = 24
Plugging these values into the formula, we get:
Area = sqrt[24(24 - 21)(24 - 17)(24 - 10)] = sqrt(24 x 3 x 7 x 14) = 84
Now we can plug in the area of the base and the height of the shape (which is given as the same for both the prism and the pyramid) into the formula for the volume of the pyramid:
Volume of pyramid = 1/3 x 84 x h
We know that the volume of the prism is 1,092 cubic units, and we can set this equal to the volume of the pyramid:
1,092 = 1/3 x 84 x h
Solving for h, we get:
h = 1,092 x 3 / (84 x 1) = 39
Now we can plug in h = 39, along with the area of the base (84), into the formula for the volume of the pyramid:
Volume of pyramid = 1/3 x 84 x 39 = 1,092 cubic units
Therefore, the volume of the pyramid is also 1,092 cubic units, option (1).
To find the total surface area of the two triangular bases of the tent, we first need to calculate the area of one triangular base and then multiply it by 2 since there are two identical triangular bases.
The formula to find the area of a triangle is:
Area = (base * height) / 2
In the given illustration, the base of the triangular face is given as 4 feet, and the perpendicular height is given as 3 feet.
Substitute these values into the formula to calculate the area of one triangular base:
Area = (4 * 3) / 2
= 12 / 2
= 6 square feet
Since we need the total surface area of both triangular bases, we need to multiply this by 2:
Total Surface Area = 6 * 2
= 12 square feet
Therefore, the total surface area of the two triangular bases of the tent is 12 square feet.