A right prism has an altitude of 16 cm and a base that is a right triangle with legs measuring 3 cm and 4 cm. Find the total surface area of a similar right prism that has a base whose legs measure 6 cm and 8 cm.
if every dimension, including the height is doubled, then the area grows by a factor of 4.
The answer is 816
To find the total surface area of a prism, we need to calculate the areas of all its faces and then add them together.
The given prism has an altitude of 16 cm and a base that is a right triangle with legs measuring 3 cm and 4 cm.
1. Calculate the area of the triangular base of the given prism:
Area = (1/2) * base * height
Area = (1/2) * 3 cm * 4 cm
Area = 6 cm²
2. Calculate the areas of the rectangular faces of the given prism:
Since it is a right prism, both rectangular faces will be congruent and have the same area.
Area of each rectangular face = base * height
Area of each rectangular face = 3 cm * 16 cm
Area of each rectangular face = 48 cm²
3. Calculate the area of the lateral faces of the given prism:
The lateral faces are formed by extending the sides of the triangular base and connecting them to form a rectangle.
The area of each lateral face will be equal to the perimeter of the base multiplied by the altitude of the prism.
Perimeter of the base = 3 cm + 4 cm + hypotenuse (using the Pythagorean theorem)
Perimeter of the base = 3 cm + 4 cm + 5 cm (since the legs of the base form a Pythagorean triplet)
Perimeter of the base = 12 cm
Area of each lateral face = perimeter of the base * altitude
Area of each lateral face = 12 cm * 16 cm
Area of each lateral face = 192 cm²
4. Calculate the total surface area of the given prism:
Total surface area = (2 * area of rectangular face) + (2 * area of lateral face) + area of triangular base
Total surface area = (2 * 48 cm²) + (2 * 192 cm²) + 6 cm²
Total surface area = 96 cm² + 384 cm² + 6 cm²
Total surface area = 486 cm²
Now, we need to find the total surface area of a similar right prism that has a base with legs measuring 6 cm and 8 cm.
5. Calculate the area of the triangular base of the similar prism:
Area = (1/2) * base * height
Area = (1/2) * 6 cm * 8 cm
Area = 24 cm²
6. Calculate the areas of the rectangular faces of the similar prism:
Area of each rectangular face = base * height
Area of each rectangular face = 6 cm * 16 cm
Area of each rectangular face = 96 cm²
7. Calculate the area of the lateral faces of the similar prism:
Perimeter of the base = 6 cm + 8 cm + hypotenuse (using the Pythagorean theorem)
Perimeter of the base = 6 cm + 8 cm + 10 cm (since the legs of the base form a Pythagorean triplet)
Perimeter of the base = 24 cm
Area of each lateral face = perimeter of the base * altitude
Area of each lateral face = 24 cm * 16 cm
Area of each lateral face = 384 cm²
8. Calculate the total surface area of the similar prism:
Total surface area = (2 * area of rectangular face) + (2 * area of lateral face) + area of triangular base
Total surface area = (2 * 96 cm²) + (2 * 384 cm²) + 24 cm²
Total surface area = 192 cm² + 768 cm² + 24 cm²
Total surface area = 984 cm²
Therefore, the total surface area of the similar right prism with a base whose legs measure 6 cm and 8 cm is 984 cm².
To find the total surface area of a similar right prism, we first need to find the lateral surface area of the given prism and then use the concept of similarity to find the lateral surface area of the similar prism.
Let's break it down step by step:
Step 1: Find the lateral surface area of the given prism.
The lateral surface area of a right prism with a triangular base can be found using the formula:
Lateral Surface Area = perimeter of the base * height of the prism
In this case, the base of the given prism is a right triangle with legs measuring 3 cm and 4 cm, so the perimeter of the base is:
Perimeter of the base = 3 + 4 + hypotenuse
Since the base is a right triangle, we can use the Pythagorean theorem to find the hypotenuse.
Hypotenuse² = 3² + 4²
Hypotenuse² = 9 + 16
Hypotenuse² = 25
Hypotenuse = √25 = 5
Therefore, the perimeter of the base is:
Perimeter of the base = 3 + 4 + 5 = 12 cm
The given prism has an altitude of 16 cm, so the lateral surface area of the given prism is:
Lateral Surface Area = 12 cm * 16 cm = 192 cm²
Step 2: Use the concept of similarity to find the lateral surface area of the similar prism.
Since the two prisms are similar, their ratios of corresponding sides are equal.
To find the ratio between the given triangle's legs and the desired triangle's legs, we divide each side of the desired triangle by the corresponding side of the given triangle.
The legs of the desired triangle measure 6 cm and 8 cm.
The legs of the given triangle measure 3 cm and 4 cm.
Therefore, the ratio of the legs is:
Ratio of legs = (6 cm / 3 cm) = 2
Ratio of legs = (8 cm / 4 cm) = 2
Since the sides of the triangles are in the ratio of 2:1, we know that the lateral surface area of the similar prism will be 2² = 4 times the lateral surface area of the given prism.
Therefore, the lateral surface area of the similar prism is:
Lateral Surface Area of similar prism = 4 * Lateral Surface Area of given prism
Lateral Surface Area of similar prism = 4 * 192 cm²
Lateral Surface Area of similar prism = 768 cm²
Therefore, the total surface area of the similar prism can be found by adding the areas of the bases to the lateral surface area:
Total Surface Area of similar prism = 2 * Base area of given prism + Lateral Surface Area of similar prism
Since the bases of the right prism are triangles, the base area of the given prism is:
Base area of given prism = (1/2) * base length * base width
Base area of given prism = (1/2) * 3 cm * 4 cm
Base area of given prism = 6 cm²
Therefore, the total surface area of the similar prism is:
Total Surface Area of similar prism = 2 * 6 cm² + 768 cm²
Total Surface Area of similar prism = 12 cm² + 768 cm²
Total Surface Area of similar prism = 780 cm²
So, the total surface area of the similar right prism is 780 cm².