Stained Glass Window Math Project

Task

The students in Mr. Rivera'southward art grade are designing a stained-glass window to hang in the school entryway. The window will exist 2 anxiety alpine and five feet wide. They accept drawn the design below:

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They take raised \$100 for the materials for the project. The colored drinking glass costs \$5 per square foot and the clear glass costs \$3 per square human foot. The materials they need to join the pieces of glass together costs 10 cents per foot and the frame costs \$4 per foot.

Do they have enough money to cover the costs of the materials they will need to brand the window?

IM Commentary

The purpose of this chore is for students to find the surface area and perimeter of geometric figures whose boundaries are segments and fractions of circles and to combine that information to calculate the cost of a project. The shape of the regions in the stained glass window are left purposefully unspecified, as one component skill of modeling with mathematics (MP4) is for students to make simplifying assumptions themselves. Given the precision needed for these estimates, bold the curves in the design are arcs of a circle is non only reasonable, it is the most expedient supposition to make too. What is important is that students recognize they are making this supposition and are explicit virtually it.

The question of whether the students have to pay for the scraps of glass left over from cutting out the shapes can exist dealt with in different means. In reality, if they had to buy the glass at a store, the glass would likely come in square or rectangular sheets and they would need to purchase more than than they were going to utilize. Merely exactly how much extra material they would take to buy depends on how the raw materials are sold, so without additional information, it would exist hard to determine that without doing some inquiry into how stained glass is sold. Alternatively, the art instructor might already accept the materials and but wants his students to stay within a sure budget for the materials they use, knowing that the scraps tin can be used for future student projects. In whatever case, this task can provide the springboard for a practiced classroom discussion around bug that students need to recall most when modeling with mathematics.

Solution

There are many means to practice this. Hither is ane:

Assume that the students only have to pay for the glass they utilise and not the scraps that they would cut away. That means we demand to figure out the area of the colored glass and the expanse of the clear drinking glass likewise as the full length of the seams betwixt the panels of drinking glass.

Showtime, we demand to detect the area of the clear glass and the area of the colored drinking glass.

The entire rectangle is 2 feet past 5 feet. Assuming that the curves are all parts of a circle with a 1 foot diameter, in that location are five one by 2 foot rectangles with either 4 half-circles or 2 half circles and 4 quarter circles of clear glass. That ways there are 2 full circles of clear glass in each 1 by 2 foot rectangle. Thus, in that location are 10 complete circles of clear glass, each with a 1 foot diameter (or a $\frac12$ foot radius). Then surface area of the entire window is 10 square feet, and the surface area of the clear glass is

$$10\times \pi (\frac12)^2= \frac52\pi$$

or approximately 7.nine foursquare feet. That means the expanse of colored glass is approximately x - 7.ix = 2.one square feet.

Now nosotros need to find the full length of the "seams" betwixt the pieces of glass.

Again, at that place are 10 circles. Their total circumference is

$$10\times \pi\times1$$

which is about 31.4 feet. At that place are besides four two-pes straight "seams." So all together there are most 39.4 feet of "seams."

The frame is two+2+5+5 = 14 feet.

The price for the clear glass is $7.ix \times 3 = 23.70$ dollars.

The cost for the colored drinking glass is $2.1 \times 5 = 10.50$ dollars.

The cost for the materials for the seams is $39.4 \times 0.10 = 3.94$ dollars.

The cost of the frame is $xiv\times4=56$ dollars.

The full cost of the materials is $23.lxx+10.l+3.94+56\approx 94$ dollars. So if these assumptions are accurate, they accept merely enough money to buy the materials. If they need to pay for the scraps or if they break pieces as they go, they don't have much wiggle room.