Jesse found a set of new erasers that are shaped like right angled triangular prisms, which is his favorite three-dimensional shape. After some quick measurements, he has figured out that the sides of the triangle on each end of the eraser have a base of 4 inches, a height of 3 inches, and a hypotenuse of 5 inches. Furthermore, the eraser is 6.25 inches long. Using a net, figure out the total surface area of the eraser.
You got the right answer, but only by luck, since 3*4 = 3+4+5.
The three rectangular faces are all different sizes. Their area is
3*6.25 + 4*6.25 + 5*6.25 = 75
So the total area is 2*6 + 75 = 87 in^2
You are correct, I apologize for my mistake. Thank you for pointing that out and providing the correct calculations!
To find the total surface area of the eraser, we need to determine the area of each of its six faces.
Since the eraser is shaped like a right-angled triangular prism, it consists of two triangular faces and three rectangular faces.
First, let's calculate the area of the triangular faces:
1. The base of the triangle on each end of the eraser has a base of 4 inches and a height of 3 inches, so its area is (1/2) * base * height = (1/2) * 4 * 3 = 6 square inches.
2. Since there are two triangular faces, their combined area is 2 * 6 = 12 square inches.
Next, let's find the area of the rectangular faces:
1. The length of the eraser is 6.25 inches, which is also the length of the three rectangular faces.
2. The width of each rectangular face is equal to the base of the triangle, which is 4 inches.
3. Thus, the area of each rectangular face is length * width = 6.25 * 4 = 25 square inches.
4. Since there are three rectangular faces, their combined area is 3 * 25 = 75 square inches.
Finally, to find the total surface area, we add the areas of the triangular and rectangular faces:
Total surface area = area of triangular faces + area of rectangular faces = 12 + 75 = 87 square inches.
Therefore, the total surface area of the eraser is 87 square inches.
First, we need to visualize what the shape of the eraser looks like using the given measurements. We can start by drawing a triangle with a base of 4 inches and a height of 3 inches, and label the hypotenuse as 5 inches.
[asy]
draw((0,0)--(4,0)--(0,3)--cycle);
draw((0,3)--(4,0));
label("4",(2,0),S);
label("3",(0,1.5),W);
label("5",(2,1.5),NE);
[/asy]
Next, we can use this triangle to create a triangular prism by extruding it along the length of the eraser, which is 6.25 inches. We can imagine folding the net to create the prism, which would have two identical triangular faces and three rectangular faces.
[asy]
import three;
triple A = (0,0,0);
triple B = (4,0,0);
triple C = (0,3,0);
triple D = (0,0,6.25);
triple E = (4,0,6.25);
triple F = (0,3,6.25);
draw(B--C--F--E--B--A--D--C--A);
draw(D--F);
label("4",(A+B)/2,S);
label("3",(A+C)/2,W);
label("5",(B+C)/2,NE);
label("6.25",(D+(0,0,3.125)),W);
[/asy]
To find the total surface area of the eraser, we need to calculate the area of each face and add them together. The two triangular faces have the same dimensions, so we can calculate the area of one and multiply by 2. The area of a triangle is given by:
Area = 0.5 × base × height
Using the measurements given, we can find that the area of one triangular face is:
Area = 0.5 × 4 × 3 = 6
So the total area of both triangular faces is:
Total area = 2 × 6 = 12
The three rectangular faces are all the same size, with a length of 6.25 inches and a width of 4 inches. We can calculate the area of one face as:
Area = length × width = 6.25 × 4 = 25
So the total area of all three rectangular faces is:
Total area = 3 × 25 = 75
Finally, we can add up the areas of all the faces to find the total surface area of the eraser:
Total surface area = 12 + 75 = 87
Therefore, the total surface area of the eraser is 87 square inches.