Question

A carpenter is making a wooden window frame that has a width of 1 inch. A window with a one-inch frame is shown. The frame is comprised of a rectangle and a semicircle. The rectangle has side lengths of 12 inches and 48 inches. The semicircle has a radius of 6 inches. The frame is 1-inch wider than the window. How much wood does the carpenter need to build the frame? 5. 5 106 square inches 5. 5 116 square inches 5. 5 153 square inches 5. 5 162 square inches.

Answer 1

You can use the fact that the middle area can be found by subtracting internal area from the total area.
The amount of wood(in area, assuming almost flat) needed by the carpenter to build the frame is given by

Option A: 5.5π + 106 sq. inches.

How to get the area between two things?

The middle area can be found by subtracting internal area from the total area.
For example, if you've got a big circle, and a small circle inside that big circle, and you want the remaining area in the big circle not occupied by smaller circle, then we have

Remaining area = Area of bigger circle - Area of smaller circle

Using the above methodology to find the amount of wood needed by the carpenter to build the given frame

The diagram is attached below.

The wooden region between those two semicircles in the top is of this below area

Area of bigger semicircle - Area of smaller semicircle.

Since bigger semicircle has 6 inch radius and smaller has 1 inch less(since width of the wood is 1 inch everywhere) which is 5 inch radius, thus,

As area of semicircle with radius r units is $\dfrac{\pi r^2}{2} \: \rm unit^2$

Thus,

Area of wood between two semicircles = $\pi \times 6^2/2 - \pi \times 5^2/2 = \pi \times 11/2 = 5.5\pi \rm \: \: inch^2$

Area of the wood in the bottom of the window = area of rectangle GLMH

Since |GL| = |HM| = 1 inch and |LM| = 12 inches, thus, as

Area of rectangle = length times width

thus,

Area of GLMH = $1 \times 12 = 12 \rm \: inch^2$

Remaining wood is as rectangle ABGE and CDFH

• Area of ABGE wood

Since we have |AL| = 48 inches but |GL| = 1 inch, thus, |AG| = 47 inches(1 less than |AL|

Also, since |GE| = 1 inch wide, thus,

Area ABGE = $1 \times 47 = 47 \: \rm inch^2$

• Area of CDFH wood

Similar to above case, we have

Since we have |DM| = 48 inches but |HM| = 1 inch, thus, |DH| = 47 inches(1 less than |DM|

Also, since |FH| = 1 inch wide, thus,

Area CDFH = $1 \times 47 = 47 \: \rm inch^2$

Thus, total wood used(in terms of area) = Area CDFG + Area ABCD + Area of bottom rectangle GLHM + Area of the internal of two semicircles on top ADCB(thick arc like shape)

or

Total area of wood = 47 + 47 + 12 + 5.5π = 5.5π + 106 sq. inches.

Thus,

The amount of wood(in area, assuming almost flat) needed by the carpenter to build the frame is given by

Option A: 5.5π + 106 sq. inches.

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