8.07 Geometric modeling

Create a model and use it to develop a plan that can be used to calculate the cost of the roof tiles for the structure.

Draw a diagram of the structure and label it with the following dimensions: b= the base length of each pyramid, p= the perpendicular height of each pyramid, r= the distance from the center of the pyramid's base to an edge, and l= the length of the flat rectangular roof between the pyramids.

A diagram with labels will help us develop a plan for calculating the area of the roof, which is needed to find the cost of the roof tiles.

A drawing of the structure with the lengths that the surveyor provides follows:

In order to calculate the cost of the roof tiles, we need the area of the roof that will be covered with the tiles. First, we will need the lateral surface area of each pentagonal pyramid, which we will assume are the same size. The lateral surface area is calculated by multiplying the area of the face of each side of the pyramid by 5.

To calculate the area of one lateral face, we need the base and height of the triangular face. Since we only know the base and the perpendicular height of the pyramid, we can use the Pythagorean theorem to calculate the height of the triangular face, as it represents the hypotenuse of the right triangle drawn from the center of the pyramid:

The length of the hypotenuse of the right triangle can be calculated as follows:

The area of each face of the pentagonal pyramids is calculated as A= \dfrac{1}{2}bh, or A=\dfrac{1}{2}b \left( \sqrt{r^2 + p^2} \right) after substituting the base and height of each triangular face. The lateral surface area of each pyramid is 5 \cdot \dfrac{1}{2}b \left( \sqrt{r^2 + p^2} \right), so the lateral surface area of both pyramids is 5b \left( \sqrt{r^2 + p^2} \right).

Then, to calculate the area of the flat rectangular roof, we can multiply the length and width of the rectangle:

The area of the flat rectangular roof is calculated as A=lw, or A=lb.

Finally, the area of the roof in square yards that will need roof tiles is represented by the following equation: S = 5b \left( \sqrt{r^2 + p^2} \right) + lb

Since the roof tiles are sold by the square foot, we will need to convert the square yards, then we can multiply the square footage of the roof by the cost of the roof tiles per square foot.

Suppose the surveyor provides the following dimensions of the parts of the structure:

• Base length of each side of the pyramids: 3 \frac{3}{4} \text{ yds}

• Perpendicular height of each pyramid: 2 \frac{1}{4} \text{ yds}

• Distance from the center of the pyramid's base to the midpoint of one edge: 2 \frac{29}{50} \text{ yds}

• Length of flat rectangular roof: 6 \frac{1}{2} \text{ yds}

If the cost per square foot of roof tile is \$9.66 / \text{sq ft}, find the total cost of the tile for the roof.

Substitute the dimensions into the equation for the area of the roof from part (a), then multiply the total area by the cost per square foot after confirming that units are aligned.

The following diagram will allow us to align the measurements to the variables we laid out in part (a):

Based on the diagram, we can note what the following variables represent:

• b= 3 \frac{3}{4}

• p= 2 \frac{1}{4}

• r= 2 \frac{29}{50}

• l= 6 \frac{1}{2}

The lateral surface area of each pyramid is calculated using the expression 5 \cdot \dfrac{1}{2}b \left( \sqrt{r^2 + p^2} \right). By subsituting the lengths of the structure, we have 5 \cdot \dfrac{1}{2} \left(3 \frac{3}{4} \right) \left( \sqrt{\left( 2 \frac{29}{50} \right)^2 + \left( 2 \frac{1}{4} \right)^2} \right)= 32.09 \text{ sq yds}.

The area of the rectangular roof is calculated using the expression lb. By substituting the lengths of the structure, we have lb = \left(6 \frac{1}{2} \right) \left( 3 \frac{3}{4} \right) = 24.38 \text{ sq yds}.

In order to calculate the cost of the roof tiles, we need to convert the areas to square feet: \text{ Pyramid roof: } 32.09 \text{ sq yds} \times \dfrac{9 \text{ sq ft}}{1 \text{ sq yd}}= 288.81 \text{ sq ft} \\ \text{ Rectangular roof: } 24.38 \text{ sq yds} \times \dfrac{9 \text{ sq ft}}{1 \text{ sq yd}}= 219.42 \text{ sq ft}

Now, we can multiply the square footage by the cost per square foot: 288.81 \text{ sq ft} \times \$9.66 \text{/sq ft} = \$2\,789.90 \\ \text{ Cost for the rectangular roof: } 219.42 \text{ sq ft} \times \$9.66 \text{/sq ft} = \$2\,119.60

The total cost of the roof tiles for the structure will be \$2\,789.90 + \$2\,789.90 + \$2\,119.60 = \$7\,699.40

Provide a report to the director of the project with the relevant information.

The report for the director of the project for the roof should be clear, so providing a labeled model of the structure will be important. We can also include a table of the areas and costs for each part of the project for thee director to make decisions about whether the roofing will be within budget, or where costs can be cut.

The cost for the roof tiles on the structure can be broken down in the following table for review:

The total cost of the tile for the roof of the structure is \$ 7 \,699.40.

If this amount is beyond the expected budget without the cost of installation taken into account, we might consider a cheaper roof tile for the rectangular part of the roof, since it is flat and customers won't even be able to see that part of the structure. Suppose we purchase a tile for the flat roof that costs \$2 per square foot, then the cost of the tile for the flat roof would decrease by over \$1\,500 to \$438.84.

As a rule of thumb, contractors will purchase more material than they actually need for a project to account for materials breaking, for human error in measuring, or based on the amount of material that can be purchased from the manufacturer. This means that the total cost would likely be higher than the given amount because of our precise measurements and because more roof tile than what is needed would likely need to be purchased.

The store manager wants each of the cones to be filled with approximately 400 \text{ cm}^3 of ice cream. Design cones that meet the given parameters.

Research on various cones will give us an idea of the dimensions of a standard sugar cone, waffle cone, or cake cone. We can start by testing out measurements we find online and determine dimensions that will give us 400 \text{ cm}^3 of ice cream.

Create a table for each ice cream cone to keep track of dimensions for the cones, then create a model with the dimensions of the cones that best fit the parameters.

For the volume of the hemispherical ice cream scoop, we will use the formula for half the volume of a sphere: \frac{1}{2}SV=\frac{1}{2} \cdot \frac{4}{3} \pi r^3, where SV is the sphere volume and r is the radius, which is half the length of the diameter of the cone.

For the volume of the ice cream in the conical and cylindrical ice cream cones, we will use the formulas for the volume of a cone and the volume of a cylinder, PV=\frac{1}{3} \pi r^2 h and CV=\pi r^2 h, respectively. PV is the pointy cone volume, CV is the cylindrical cone volume, and r is the radius of each cone.

Since some conical-shaped ice cream cones have a diameter of about 5 \text{ cm} and a height of about 11 \text{ cm}, we can use those dimensions as our starting point.

By using an ice cream scoop with a diameter of 7 \text{ in}, we will use the same diameter for the flat-based cone as follows:

By starting with ice cream cone dimensions that are standard sizes and making adjustments, we can calculate the volume of ice cream needed. A model for the ice cream cones is shown below:

The parameters for the ice cream cones are given, so we have the essential features of the problem. While a diagram is shown, the model we created for the problem must have actual dimensions. A tabular model also helped us to sort out information and possibly look for potential patterns in how the volume changes as the scoop and cone change.

Note that there are other approaches to the problem, and other choices we could make for radius and height to meet the volume requirements.

Consider the dimensions you chose for the cones you created in part (a). Think of what it would be like to hold each cone in your hands. Are your dimensions practical?

One way to interpret the practicality of our model is to measure out and create a physical model of the cones. We can do this using scissors, paper, and a ruler. Since both hemispheres on top of the cones are the same, we can just examine the cone sizes.

With a physical model, we can discuss the practicality or relate the model to our experiences. With our models, the cylindrical flat-based cone feels more like a cup that ice cream would be served in. The pointy cone needs to be very large to fit the ice cream needed, and it has the feel of a waffle cone.

Interpreting our models helps us to make better decisions about the math and determine how we will improve or report our findings.

Provide a report to the store manager with recommendations for the dimensions of the cones if they want each to have approximately 400 \text{ cm}^3 of ice cream in each. Include both practical and mathematical considerations, and make any suggestions you think might be helpful for the manager's plans.

The following ice cream containers can be offered to customers with approximately 400 \text{ cm}^3 of ice cream in each:

Due to the sizes of the cones, we can refer to the cylindrical cone as a bowl and continue to call the conical cone a cone. The bowl holds approximately 411 \text{ cm}^3 of ice cream while the cone holds 385 \text{ cm}^3.

By calling our cylindrical vessel a bowl instead of a cone, we can market this option at a different cost than the cone. Even though both vessels will hold approximately the same amount of ice cream, we can make a profit based on how much ice cream is perceived to be in each container.

After interpreting the solutions from our model, our report should also be practical. In this situation, we are creating two containers that hold the same amount of ice cream. The purpose for the manager should be to improve their business in some way, so just offering a variety of containers that will earn the same amount of money may not be worth the effort.

There could be other valid recommendations that could be made in the report. For example, we could have recommended that the manager create a cylindrical cone with a longer height and more volume, so it looks more like a cone that customers expect. Then, charge a price for both cones that covers the cylindrical cone's volume, increasing the profit on each conical cone sold.