A jeweler has three solid gold cylinders with radii 2.0 millimeters, 3.0 millimeters, and 4.0 millimeters. Each cylinder
has a height of 10.0 millimeters. If the jeweler melts the cylinders down to make a solid prism with a square base of
length 5.0 millimeters, what will be the approximate height of the prism? Round to the nearest tenth.
the total volumes of the cylinders must equal the volume of the prism, so
π(2^2 + 3^2 + 4^2)*10 = 5^2 * h
To find the approximate height of the prism, we need to calculate the volume of the gold cylinders and then divide it by the base area of the prism.
First, let's calculate the volume of each cylinder using the formula V = πr^2h, where r is the radius and h is the height:
Volume of cylinder 1: V1 = π(2.0 mm)^2 * 10.0 mm
Volume of cylinder 2: V2 = π(3.0 mm)^2 * 10.0 mm
Volume of cylinder 3: V3 = π(4.0 mm)^2 * 10.0 mm
Next, we can find the total volume of the three cylinders by adding them together:
Total volume of cylinders: V_total = V1 + V2 + V3
Now, let's calculate the base area of the prism using the formula A = side^2, where side is the length of the square base:
Base area of prism: A_base = (5.0 mm)^2
Finally, we can find the height of the prism by dividing the total volume of the cylinders by the base area of the prism:
Height of prism: h_prism = V_total / A_base
Now, let's calculate the values:
Volume of cylinder 1: V1 = π(2.0 mm)^2 * 10.0 mm ≈ 125.7 mm^3
Volume of cylinder 2: V2 = π(3.0 mm)^2 * 10.0 mm ≈ 282.7 mm^3
Volume of cylinder 3: V3 = π(4.0 mm)^2 * 10.0 mm ≈ 502.7 mm^3
Total volume of cylinders: V_total = V1 + V2 + V3 ≈ 125.7 mm^3 + 282.7 mm^3 + 502.7 mm^3 ≈ 911.1 mm^3
Base area of prism: A_base = (5.0 mm)^2 = 25.0 mm^2
Height of prism: h_prism = V_total / A_base ≈ 911.1 mm^3 / 25.0 mm^2 ≈ 36.44 mm
Rounding to the nearest tenth, the approximate height of the prism will be 36.4 mm.
To find the approximate height of the prism, we can start by calculating the volume of each cylinder and then adding them together. Since the cylinders have the same height, the sum of their volumes will be equal to the volume of the prism.
The formula for the volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.
For the first cylinder with a radius of 2.0 millimeters, the volume is V1 = π(2.0^2)(10.0) ≈ 125.7 cubic millimeters.
For the second cylinder with a radius of 3.0 millimeters, the volume is V2 = π(3.0^2)(10.0) ≈ 282.7 cubic millimeters.
And for the third cylinder with a radius of 4.0 millimeters, the volume is V3 = π(4.0^2)(10.0) ≈ 502.7 cubic millimeters.
Now, we can add the volumes of the three cylinders to get the total volume of the prism: V_prism = V1 + V2 + V3.
V_prism = 125.7 + 282.7 + 502.7 ≈ 911.1 cubic millimeters.
To find the approximate height of the prism, we can use the formula for the volume of a prism: V_prism = base area × height. In this case, the base of the prism is a square with a side length of 5.0 millimeters.
The volume of the prism is V_prism = (5.0^2)(height) = 25.0 × height.
We can set this equal to the calculated volume of the prism and solve for the height: 25.0 × height = 911.1.
Dividing both sides of the equation by 25.0, we get height = 911.1 / 25.0 ≈ 36.4.
Thus, the approximate height of the prism will be 36.4 millimeters (rounded to the nearest tenth).