8.05 Modeling with triangles

Adalberon reads information regarding advice and safety measures for the rise and run of each stair. The angle of elevation from one stair to the next can vary from 30 \degree to 50 \degree. Given that the run of each stair is 11 \text{ in}, create a model with an appropriate rise for each stair.

Crate a model of a staircase, with the lengths of the treads, rise of the stairs, and angle of elevation labeled.

Assuming that Adalberon thinks the best plan is to construct a staircase using the average angle of elevation between each tread, let the angle of elevation \theta = 40 \degree.

The following equation can be set up to find the necessary height of each riser on the stairs, given that the angle of elevation is 40 \degree: \tan 40 = \dfrac{x}{11}

After multiplying each side of the equation by 11, we have the solution x=9.23. This means that the height of each riser on the stairs should be 9.23 \text{ in}. Since 9.23 \text{ in} is close to 9.25 \text{ in}, Adalberon would be able to use the 9 \frac{1}{4} \text{ in} marking on a measuring tool to cut each riser, as shown:

We have to start a problem by making assumptions. The assumption here is that Adalberon is going to just choose a number between the suggested angles of elevation he found.

Adalberon's parents think he should make the angle of elevation between 30 \degree and 35 \degree for his grandparents' safety. They also tell Adalberon to consider cutting new treads for the stairs so that he is not restricted to a specific run for each stair and that he use an official website that will have safety guidelines for stairs.

The Occupational Safety and Health Administration (OSHA) in the United States Department of Labor has requirements that Adalberon can use for the deck stairs:

• The rise of each stair should be between 6 \text{ in} and 7 \frac{1}{2} \text{ in}

• The run of each stair should be between 10 \text{ in} and 14 \text{ in}

Revise your model from part (a) and apply the new requirements.

For the new model, Adalberon could focus on the OSHA standards for riser and tread lengths, and then calculate the resulting angles of elevation. Create a new model to compare the possible tread lengths and riser heights to meet the comfort and safety guidelines.

The following equation can be used and evaluated with various tread lengths and riser heights for Adalberon to make a decision on the right combination:

A table is a useful model for organizing combinations of tread lengths and riser heights:

Using just these combinations, stairs with a height of 7 \text{ in} and a length of 11 \text{ in} will have an angle of elevation between 30 \degree and 35 \degree, making the combination safe and comfortable.

The deck sits 4 \frac{1}{2} \text{ ft} above the ground. Analyze your new model from part (b) and use it to plan out a report for Adalberon's parents that includes the model, the dimensions of each stair and the stairway as a whole, and a rationale for why this plan will meet all the necessary criteria.

We need to determine how many treads and risers are necessary to meet the distance of the deck to the ground. The deck stands at 4 \frac{1}{2} \text{ ft}, or 54 \text{ in}, so a total of 54 \text{ in} \div \dfrac{ 7 \text{ in}}{1 \text{ riser}} = 7.7 or 8 risers are required for the staircase. So, 54 \text{ in} \div {8 \text{ risers}} = 6.75. That means instead of 7-inch risers, the risers should be 6.75 \text{ in}. With 11 \text{ in} treads, the angle of elevation will be 31.5 \degree, so still within the range needed.

That means Adalberon needs 7 treads. A new model showing some of the necessary dimensions will help Adalberon make sense of the materials he needs cut and help him check that his math works:

The 7 treads multiplied by 11 \text{ in} each gives us a horizontal distance into the yard of 77 \text{ in} \times \dfrac{1 \text{ ft}}{12 \text{ in}} = 6 \text{ ft } 5 \text{ in}. By using the Pythagorean theorem, we can calculate the distance from the bottom of the stairs to the top as follows:

The distance from the top to the bottom of the staircase is about 7.84 \text{ ft}, which we can round to about 7 \text{ ft }10 \text{ in}.

Analyzing the model with different factors in mind (e.g. angle of elevation for each stair, rise and run requirements, total vertical height needed) may prompt us to continually revise the model.

If 1 acre is equivalent to 43 \, 560 \text{ ft}^2, create a model of the plot of land if Florina only chooses to sell the minimum required road frontage of 120 \text{ ft}. Determine whether this is a good model and consider the conclusions of selling the plot of land using this model.

Using the surveyor's diagram, we can mark the image with variables to represent various lengths and angles, and make sure that the road frontage that Florina is selling is only 120 \text{ ft}.

Note: Diagram not drawn to scale

The model can be used with the formula for finding the area of non-right triangles to start solving for the missing dimensions of the plot of land:

Now, we can use the law of cosines to calculate the length of the property that Florina needs to purchase a fence for:

The side of the property that Florina needs to purchase fencing for is 1\,113.82 \text{ ft} long.

Since 2 \text{ yds} is equivalent to 6 \text{ ft}, we can divide 1\,113.82 \text{ ft} by 6 \text{ ft} to determine the number of panels Florina needs to buy, and then multiply the number of panels by \$70: 1\,113.82 \text{ ft} \div 6 \text{ ft} = 185.64 \\ 186 \text{ panels needed} \times \$70 = \$13 \,020

If Florina decides to sell this plot of land, she will benefit by keeping most of the road frontage on her property. However, selling only 120 \text{ ft} of road frontage may not be as easy to sell because it may not be appealing to buyers. Also, the property line for which she has to purchase a fence will be quite long, and therefore expensive.

While this serves as a potential model for how much land Florina is going to sell, we have no other models to compare it to, and there are factors that might not yield a fast sale of the land or may cost Florina more money than she may need to spend on the fence.

After creating, applying and analyzing our model, we may need to adjust the model to provide more complete information to make an informed decision.

Apply the concept from part (a) to revise your model and create a new model that will help Florina compare different-sized road frontage options.

We can use the diagram to construct a table of values that includes the possible lengths of road frontage and the length of the property line for which she would need to purchase fencing. Since Florina has a range of distances along the road that she is willing to sell, we should consider the different configurations that she can partition off of her land.

Using the same approach as part (a), we will start with the road front length as b, use the formula for the area of a non-right triangle to find the length, a, on the back of the property, then use the law of cosines to find the length of the shared property line:

As Florina opens the possibility of selling more of the roadside of her land, she also decreases the length of the land that she needs to purchase fencing for. With a table that models the various options for selling her land and purchasing fencing, Florina can make a more informed choice.

Interpreting our revised model verifies that we have sufficient information about the various important parameters in the problem (road frontage, property dimensions, cost of fencing) so we can make an informed decision.

Make a final recommendation to Florina about the dimensions of the parcel she should sell.

After offering Florina some options, it will be important to consider the factors that will influence her decision about the partition of land she will sell, including the cost of fencing, the desirability of the property based on its shape and road frontage, and how much road frontage to keep on her property.

The limited parameters of road frontage that Florina is willing to part ways with will make the sale of a portion of her property challenging. In general, the more road frontage that Florina is willing to sell, she will spend less money now and potentially find a buyer faster.

Comparatively, if Florina sells property with 120 \text{ ft} of road front, the triangular-shaped plot will be extremely narrow and long. The length of fence Florina will need to purchase is about 1\,150 \text{ ft} long for \$13\,000. However, if Florina sells property with 200 \text{ ft} of road front, the plot will be less narrow. The length of fence Florina will need to purchase is about 650 \text{ ft} long for \$7\,600.

By offering more road frontage, Florina will offer a more desirable plot of land and the fence will cost her almost half as much as less road frontage in the sale.

Once we have all the information, we need to report on our recommendations, making sure to include the assumptions and reasoning that led to the decision.

Find the approximate area covered by the amphitheater to the nearest square meter.

Using the description of the amphitheater, we can approximate the area to that of a composite figure formed by a rectangle and a semicircle.

Area of the semicircle:

Area of the rectangle:

Area of the amphitheater:

If we evaluate the expression for the area of the amphitheater and round to the nearest square meter, we get that:A\approx 277\text{ m}^2

If there are 300 people in the audience and 20 actors on stage, find the population density of the amphitheater.

Rounding to two decimal places, the population density of the amphitheater is 1.16 people per square meter.

Consider the following:

• How are the units determined for a population density?

• How the population of the stage compares to that of the audience section?

• Why might population density at a venue be important?

The units for a population density come from the units of the population divided by the units of the area. In the example above we had a population of individual people and an area in square meters. Thus, the units were "people per square meter". Other examples could include units such at "hundreds of cats per square mile" or "mites per square inch".

The population density of the stage, 0.08 people per square meter, is significantly lower than that of the audience section, 1.91 people per square meter.

Population density calculations can play important roles in regulations such as fire safety, or restrictions during a pandemic.