10.03 Prisms and cylinders

To solve this, we will use the formula for the volume of a cylinder with the given radius and height lengths.

We have been given the values for the radius, r=5, and the height, h=13, so we can substitute these values into the volume formula.

As an exact value, this would be 325\pi \text{ cm}^3.

To create the pipe, we must find the difference in the area of the outer circle and the inner circle. Once we have found the difference in the area of the circles, we can use that area to find the volume using the formula V=Bh.

To find the base area, subtract the area of the inner circle from the outer circle. For this problem, we will let the base area B=A_{\text{base}}.

The volume of concrete required is 816.81\, \text{ cm}^3.

We can identify the volume in one set of workings:

This can help with accuracy, because if we had rounded the area of the base to 20\pi=62.83 \text{ cm}^2, then our answer would be V=62.83\cdot 13=816.79 \text{ cm}^3.

Being off by 0.02 sometimes does not make a difference, but if we make that rounding error for 1 \,000\,000 pipes, then it can add up to be significant.

To find the volume, we must first find the area of the triangle. Once we have the area of the triangle, we can use the volume of a right prism formula, V=Bh.

This figure is composed of a rectangular prism and a half cylinder. First, we will find the volume of each shape and then add the two volumes together.

The half cylinder has a diameter of 4\text{ m} which means its radius is 2\text{ m}. The rectangular prism has a side length of 4\text{ m} and a height of 10\text{ m}.

Since this the volume of a full cylinder, we need to divide this value by 2 to find the volume of half of it.125.66 \div 2 = 62.83 \text{ m}^3Next, we will find the volume of the rectangular prism:

An alternative solution would be to find the area of the base of this solid and then multiply it by the height of the solid.

We can use the volume of a right prism formula, V=Bh, to substitute our known values for the volume, 990 \text{ cm}^3, and the base area, 110 \text{ cm}^2. Then, we will solve for the missing value of h.

We know that only the height is going to change, which means the base area will stay the same. We can use the volume of a prism formula, V = Bh, where B is the base area and h is the height.

If the new volume is 27 times greater, then V_{\text{new}} = 27\cdot V=27 \cdot Bh As the new cube has the same length and width, the only dimension that can change to increase the volume is the height.

That means the height of the new box must be 27 times the height of the original box to achieve the new volume.

Add the areas of the five faces of the prism, where the top and bottom face are identical:

Because the top and bottom face are congruent, we can find twice the area of one face. Notice that the top and bottom are right triangles, so the base and height will be the lengths of each leg.

We need to determine whether the entire top of each layer needs to be iced, and then the next layer added, or if the layers are placed first, and then only the exposed portions are iced.

After finding the surface area, we will want to multiply that value by \dfrac{1}{8} \text{ inches}. Depending on the approach to the problem, we could end up with different solutions to how much icing Yuki needs for the cake.

Let's assume that the layers are placed, and then only exposed cake will be iced. That means we should calculate the lateral area of each layer, then calculate the tops of the tiers shown in the diagram using the areas of the circular tops.

For the lateral area of each tier, we need the circumference of each.

Starting with the first tier we have C=2 \pi r \to C = 2 \pi \left(2 \text{ in} \right) \to C = 12.57 \text{ in}

So, the lateral area of the first tier is 12.57 \text{ in} \cdot 2 \text{ in}= 25.14 \text{ in}^2.

For the middle tier we have C=2 \pi r \to C = 2 \pi \left(4 \text{ in} \right) \to C = 25.13 \text{ in}

So, the lateral area of the middle tier is 25.13 \text{ in} \cdot 3 \text{ in}= 75.39 \text{ in}^2.

For the bottom tier we have C=2 \pi r \to C = 2 \pi \left(6 \text{ in} \right) \to C = 37.7 \text{ in}

So, the lateral area of the bottom tier is 37.7 \text{ in} \cdot 4 \text{ in}= 150.8 \text{ in}^2.

The top of the first tier will be iced, so we can find the area of the circular top. A=\pi r^2 \to A = \pi \left(2 \text{ in} \right)^2 = 12.57 \text{ in}^2

The exposed portion of the middle layer can be found by subtracting the area of the first tier's base from the area of the middle tier's base. The area of the top of the middle tier can be calculated as A=\pi r^2 \to A = \pi \left(4 \text{ in} \right)^2 =50.27 \text{ in}^2 Then, we can subtract the area of the first tier's base 50.27 \text{ in}^2 - 12.57 \text{ in}^2 = 37.7 \text{ in}^2

The exposed portion of the bottom layer can be found by subtracting the area of the middle tier's base from the area of the bottom tier's base. The area of the top of the bottom tier can be calculated as A=\pi r^2 \to A = \pi \left(6 \text{ in} \right)^2 =113.1 \text{ in}^2 Then, we can subtract the area of the middle tier's base 113.1 \text{ in}^2 - 50.27 \text{ in}^2 = 62.83 \text{ in}^2

Finally, we will add the total surface area of icing needed for the cake and multiply the total by \dfrac{1}{8} \text{ in} to find the amount of icing needed for Yuki's cake:25.14 \text{ in}^2 + 75.39 \text{ in}^2 + 150.8 \text{ in}^2 + 12.57 \text{ in}^2 + 37.7 \text{ in}^2 + 62.83 \text{ in}^2 = 364.43 \text{ in}^2 \\ 364.43 \text{ in}^2 \cdot \dfrac{1}{8} \text{ in} = 45.55 \text{ in}^3