8.1E Exercises

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Note: Figures may not be drawn to scale! Yes, I'm a talented artiste, but I am also lazyyyy.

Also note: When numbers are "well cooked" (small and whole numbers) don't use a calculator. If dimensions have fractions or decimals or you need to use the Pythagorean Theorem and it gets ugly, you can use a calculator.

Rectangles

Find the area and perimeter.

$b1.png$	$b2.png$	$b3.png$	$b4.png$
1.	2.	3.	4.

Answer

$A = 25$ sq units. $P = 20$ units.
$A = 51$ sq units. $P = 40$ units.
$A = 2a^2$ sq units. $P = 6a$ units.
$ A = (a+b)(a-b) = a^2 - b^2 $ sq units. $ P = 2(a+b) + 2(a-b) = 4a $ units.

Triangles

Find the area and perimeter.

$a1.png$	$a2.png$	$a3.png$	$a4.png$
1.	2.	3.	4.

Answer

$A = 6$ sq units. $P = 12$ units.
$A = 60$ sq units. $P = 36$ units.
$A = \sqrt{3}x^2$ sq units. $P = 6x$ units.
$A = 2.365$ sq units. $P = 7.18$ units.

Circles

Give the diameter, circumference, and area.

$b5.png$	$b6.png$	$b7.png$	$b8.png$
1.	2.	3.	4.

Answer

$d = 14$ units, $C = 14\pi$ units, $A = 49\pi$ sq units
$d = 22$ units, $C = 22 \pi$ units, $A = 121 \pi$ sq units
$ d = 854$ mm, $C = 854\pi$ mm, $A = 182329 \pi$ sq mm.
$ d = 6w$ units, $C = 6 \pi w$ units, $A = 9\pi w^2 $ sq units.

Composite Shapes

Find the area and perimeter.

$c1.png$	$c2.png$
1.	2.
$c3.png$	$c4.png$
3.	4.

Answer

$A = 48$ sq units. $P = 8+16+2(4\sqrt{2}) = 24+8\sqrt{2} $ units.
$A = 88$ sq units. $P = 36$ units.
$A = 5 \sqrt{3}$ sq units. $P = 10$ units.
$A = 4\pi - \pi = 3\pi$ sq units. Perimeter is the inner circumference plus the outer circumference, $P = 4\pi + 2\pi = 6\pi$ units.

Boxes

Find the volume and surface area.

$c5.png$	$c6.png$
1.	2.
$c7.png$	$c8.png$
3.	4.

Answer

$V = 320$ cu units. $S = 288$ sq units.
$ V = abc$ cu units. $S = 2ab + 2bc + 2ac$ sq units.
$ V = 120h$ cu units. $S = 240+44h$ sq units.
$ V = (x+y)^2(x-y) = x^3 +x^2y - xy^2 - y^3$ cu units. $S = 2(x+y)^2 + 4(x+y)(x-y) = 6x^2 + 4xy - 2y^2$ sq units.

Extruded Solids

Find the volume and surface area.

$d1.png$	$d2.png$	$d3.png$
1.	2.	3.

Answer

$V = 720\pi$ ft$^3$. $S = 2(36\pi)+20(12\pi) = 312\pi$ ft$^2$.
This is a hollow shape, the same size as the previous cylinder, but subtracting an inner cylinder with radius 5ft and length 20ft. The volume of the interior cylinder is $500\pi$, so the volume of the shell shape is $V = 720\pi - 500\pi = 220\pi$ ft$^3$. The surface area is tricky. There is the outer wall, $20(12\pi)$, and the inner wall, $20(10\pi)$, and the front and back surfaces both have area $ 30\pi - 25\pi$. All together, it's $S = 450\pi$ ft$^2$.
$V = 6$ cubic units. $S = 16 + 6\sqrt{2}$ square units.

Spheres

Find the volume and surface area.

$d4.png$	$d5.png$	$d6.png$
1.	2.	3.

Answer

$V = 36\pi$ cubic units. $S = 36\pi$ square units.
$V = \frac{4}{3}R^3 \pi$ cubic units. $S = 4\pi R^2 $ square units.
$ V = \frac{4}{3}\pi$ cubic inches. $S = 4\pi$ square inches.

Lassoing the Earth

I have 25,000 miles worth of rope lying around so I decide to wrap it all the way around the Earth, right on the equator. Then I decide I want to raise the rope 1 meter above the ground everywhere, all the way around. How much more rope do I need? Pause first and intuitively make a guess...the answer might surprise you!

Answer: Hint: The rope forms a circle which has some radius. How does the radius change?

Square Pies?

My sister and her husband are coming to visit me for Thanksgiving and I plan to make two pumpkin pies (the correct choice of pie). The recipe on the can of pumpkin says it makes two 9-in pies. However, I only have one traditional round 9-inch-diameter pie pan. I do have several other sizes of rectangular glass baking dishes that I could frankenstein a pie crust into, though...

An 8-inch by 8-inch square
A 7-inch by 11-inch rectangle
A 9-inch by 13-inch rectangle

Compare the areas of all the shapes to see which of the three I should use in addition to my standard pie pan. (You can use a calculator.)

Answer: Find the area of the pie pan by finding its radius first. The 8x8 pan is the best approximation of the circular pan's area.

My Brother's Pantry

As mentioned in the section, my brother bought a house. He actually really truly literally called to have me help solve the following problem while installing pantry shelving, since the house was built in the 60s and has wonky dimensions. You can use a calculator on this problem, and although figures are not drawn to scale, you can assume angles that look like right angles are right angles.

Step 1: Here are the dimensions he was able to measure, since it was difficult physically to get accurate measurements inside the pantry. Use the given dimensions on the triangles to find the length "?" below. Round your answer to four decimal places.

$pantry1.png$

Step 2: Plug the number you got in Step 1 into the blue dimension "?" below, and find the length labeled "??" rounded to four decimal places.

$pantry2.png$

Step 3: My brother uses a standard tape measure while cutting his boards, on which the smallest tick marks are $ \dfrac{1}{16}$ of an inch. Find a way to write your decimal approximation for "??" as a number he can see easily on the tape measure.

Answer

Step 1: Using the Pythagorean Theorem on the black triangle and adding appropriately, you should get ? = 23.7234 inches.

Step 2: You should get ?? = 23.6759 inches.

Step 3: Using my calculator to find decimal value conversions of fractions with denominator 16, I find that $0.6759$ is just a hair under $\frac{11}{16} = 0.6875$. This tick mark would be the small one right after a $ \frac{5}{8}$-ths tick. He should measure the board to be $23 \frac{11}{16}$ inches.

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