Sectors, Areas, and Arcs: Word Problems

This question is actually fairly straightforward. All I need to do is understand that they've given me a radius (being the value for "everything within") and an angle measure, and they've asked me to find the area of the resulting sector. All I have to do is plug this info into the sector-area formula (remembering to use a conversion factor in order to convert to radians), and simplify:

For my final answer, I'll need to round, and I'll need to include the proper units.

In essence, they've given me the central angle of a sector and that sector's arc's length, and they've asked me for the radius. So I'll plug into the arc-length formula, and solve for what I need. (In this case, I won't need to use a conversion factor, because I can use the radian form for "two-thirds of a circle". But I will want to convert the mixed number to an improper fraction.)

The units for this question are "feet", so my answer is:

The swing arm for the wiper can be viewed as being the radius of a circle whose center is at the pivot point. No, the swing arm won't rotate through a complete circle, but I don't care; I only need the fact of the circle, because this allows me to use the circle-sector formulas. In this case, they've given me the radius and the subtended angle, and they want me to find the area, so I'll be using the sector-area formula. However, the wiper blade itself does not go from the tip of the swing arm, all the way down to the pivot point; it stops short of the pivot point (or, in this mathematical context, the center of the circle). Because of this, I'll need to subtract a portion of the entire sector (created by the swing arm) to find the area of the windshield that is actually covered by the blade.

Using the entire length of the swing arm as my radius, I get the area of the swing-arm's sector (using the conversion factor to swap radians for degrees) as being:

I have to remember that this is the total area swept by the swing arm. The wiper blade only covers the outer 60 cm of the length of the swing arm, so the inner 72 − 60 = 12 centimeters is not covered by the blade. This inner area, forming a sector of a smaller circle, needs to be subtracted from the previous value, in order to find the area that is swept by the actual wiper blade. This smaller area, using the same angle measure but having a smaller radius (namely, 12 cm), is:

Note how I have left my areas in "exact" form. I don't want to create problems with round-off error, so I'll wait to round "to the nearest square centimeter" until the end. At this point, I need to subtract the area I just found from the first (larger) area:

Remembering the units and that I'm supposed to round to "the nearest square centimeter" (in this context, to the nearest whole number), my answer is:

This is a lot of information, but all I need are the geometrical parts. Because half a centimeter is added to every side of the fabric, the width of the bolt of cloth (in other words, the width of the long rectangle above) must be twenty-two centimeters. This must also be the radius of each of the sectors which represents each fan's fabric.

So if I can find the area of the rectangle between the top and bottom solid black lines and the left and right dashed red lines, and subtract from this the (green) area of the fans' fabrics, I'll have the area of the (white) "waste" fabric.

Taking a good look at the picture, I see that what the die cuts with each revolution includes the last portion of one right-side-up blank, an entire upside-down blank, and the first portion of another right-side-up blank. Together, these three areas are equal to two complete fan blanks. So I can find the shaded-green area by finding the area of one sector (remembering to convert to radians), and multiplying the result by 2.

(I'm leaving everying in "exact" form until much closer to the end, to avoid round-off error.)

Now that I have the area of two fans, I need the area of the whole rectangle over which the cutting die is passing. (This is the easy part!)

The "waste" is the white part that's left over after they remove the part of the silk that they want (being the shaded-green blanks or cutouts), so I can subtract to find this value. It is after this point (because I'm doing the computations in my calculator) that I'll do the rounding:

I need to round this to one decimal place, and I need to remember that the units are centimeters and that this is an area, so my answer is:

Okay; this looks impossible, but that's mostly because they gave me only letters. I have no actual numbers that I can use. But I have two formulas, so I'll fiddle around with that, and see where it takes me. Plugging what I have into the two formulas, I get:

What happens if I plug the new radius value into the first line above (the one for the arc length) in place of the "r"?

This is twice the original expression, so I can see that the new arc length is equal to 2L; that is, the new arc length is twice the original arc length. So what if I do the same thing in the second equation?

In the first parenthetical, I've got the original expression for the original area. In the second parenthetical, I've got a 4. This tells me that the new sector area is four times as big as the original sector area. Since they haven't given me any numbers to work with, I'll assume that these comparisons are what they're wanting as an answer.