An Art Student Wants to Make a Model of a Rectangular Classroom

Stained Drinking glass

Task

The students in Mr. Rivera'south art class are designing a stained-glass window to hang in the schoolhouse entryway. The window will exist 2 feet tall and 5 feet broad. They have drawn the blueprint below:

They accept raised \$100 for the materials for the project. The colored glass costs \$5 per square human foot and the clear glass costs \$three per square 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 task is for students to notice 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, every bit 1 component skill of modeling with mathematics (MP4) is for students to make simplifying assumptions themselves. Given the precision needed for these estimates, assuming the curves in the design are arcs of a circle is non only reasonable, it is the nearly expedient assumption to make as well. What is important is that students recognize they are making this supposition and are explicit about it.

The question of whether the students have to pay for the scraps of glass left over from cutting out the shapes can be dealt with in different ways. In reality, if they had to buy the drinking glass at a store, the glass would likely come in square or rectangular sheets and they would need to buy more than they were going to use. But exactly how much extra material they would have to buy depends on how the raw materials are sold, and then without boosted information, information technology would be difficult to decide that without doing some inquiry into how stained glass is sold. Alternatively, the art instructor might already accept the materials and just wants his students to stay within a certain budget for the materials they use, knowing that the scraps tin be used for future educatee projects. In any case, this job tin provide the springboard for a proficient classroom give-and-take around issues that students need to call back virtually when modeling with mathematics.

Solution

There are many ways to do this. Here is one:

Assume that the students merely take to pay for the glass they use and non the scraps that they would cut away. That means we need to effigy out the area of the colored glass and the area of the articulate drinking glass likewise as the total length of the seams between the panels of drinking glass.

First, we demand to notice the area of the clear glass and the area of the colored glass.

The unabridged rectangle is two feet past 5 feet. Assuming that the curves are all parts of a circle with a 1 foot diameter, there are five 1 past 2 foot rectangles with either 4 half-circles or ii one-half circles and 4 quarter circles of clear drinking glass. That means there are 2 full circles of clear drinking glass in each 1 by 2 pes rectangle. Thus, there are 10 consummate circles of clear glass, each with a 1 foot diameter (or a $\frac12$ foot radius). So area of the unabridged window is 10 square anxiety, and the area of the clear glass is

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

or approximately seven.ix square feet. That means the area of colored glass is approximately ten - 7.9 = 2.1 square feet.

Now we need to notice the full length of the "seams" between the pieces of glass.

Again, there are 10 circles. Their full circumference is

$$x\times \pi\times1$$

which is about 31.4 anxiety. There are also four two-foot straight "seams." So all together there are virtually 39.4 feet of "seams."

The frame is 2+2+5+5 = 14 anxiety.

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

The toll for the colored glass is $2.1 \times five = x.fifty$ dollars.

The toll for the materials for the seams is $39.four \times 0.10 = 3.94$ dollars.

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

The total cost of the materials is $23.seventy+10.50+3.94+56\approx 94$ dollars. Then if these assumptions are accurate, they have just 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.

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