You are choosing between two different cheese wedges at the grocery store. Assume both wedges are triangular prisms with bases that are isosceles triangles. The first wedge has a base that is 2.5 in. wide and a height of 4.5 in., with the entire wedge being 3 inthick. The second wedge has a base that is 3 in. wide and a height of 4 in., with the entire wedge being 3.5 in. thick. Which wedge has a greater volume of cheese, and by how much?
The first wedge by 6.375 cubic inches
The first wedge by 12.75 cubic inches
The second wedge by 8.25 cubic inches
The second wedge by 4.125
So what is the answer and why?
To determine which cheese wedge has a greater volume, we can use the formula for the volume of a triangular prism. The formula is V = (1/2) * base * height * thickness.
For the first cheese wedge:
Base = 2.5 in.
Height = 4.5 in.
Thickness = 3 in.
So, the volume of the first wedge is V1 = (1/2) * 2.5 * 4.5 * 3 = 16.875 cubic inches.
For the second cheese wedge:
Base = 3 in.
Height = 4 in.
Thickness = 3.5 in.
So, the volume of the second wedge is V2 = (1/2) * 3 * 4 * 3.5 = 21 cubic inches.
To find the difference in volume, we subtract V1 from V2: V2 - V1 = 21 - 16.875 = 4.125 cubic inches.
Therefore, the second wedge has a greater volume of cheese than the first wedge, by 4.125 cubic inches.
To determine which cheese wedge has a greater volume, we need to calculate the volume of each wedge and compare them.
The volume of a triangular prism can be calculated using the formula: V = (base area) x height.
Let's calculate the volume of each wedge:
First wedge:
Base width = 2.5 in.
Height = 4.5 in.
Thickness = 3 in.
To find the base area of the first wedge, we need to calculate the area of the isosceles triangle.
The formula for the area of an isosceles triangle with base b and height h is: A = (b x h) / 2.
In this case, the base of the isosceles triangle is the width of the cheese wedge (2.5 in) and the height is the height of the cheese wedge (4.5 in).
So, the base area of the first wedge is: A = (2.5 x 4.5) / 2 = 5.625 square inches.
Now, let's calculate the volume of the first wedge:
V1 = (base area) x height = 5.625 sq in x 3 in = 16.875 cubic inches.
Second wedge:
Base width = 3 in.
Height = 4 in.
Thickness = 3.5 in.
Similarly, let's find the base area of the second wedge:
A = (base width x height) / 2 = (3 x 4) / 2 = 6 square inches.
Now, let's calculate the volume of the second wedge:
V2 = (base area) x height = 6 sq in x 3.5 in = 21 cubic inches.
Comparing the volumes, we find that the first wedge has a volume of 16.875 cubic inches, while the second wedge has a volume of 21 cubic inches.
Therefore, the second wedge has a greater volume by: 21 - 16.875 = 4.125 cubic inches.
So, the correct answer is: The second wedge by 4.125 cubic inches.
I assume you can find the area of a triangle. That is the base area B.
As with all prisms, V = Bh
so plug in your numbers and compare.