Q: A window consists of a rectangle 4 ft wide and 6 ft tall, capped by a semicircle. What is the total perimeter of the window (to the nearest foot)? Use $$\pi \approx 3.14.$$

Perimeter $$= 2 \times 6 + 4 + (\pi \times 2) = 18 + 6.28 = 24.28 ft$$

Q: A triangle has sides of 5 cm, 5 cm, and 6 cm, with a height of 4 cm to the base of 6 cm. What is its area?

$$A = \frac{1}{2} \times 6 \times 4 = 12 cm$^2$$

Q: A rectangular garden is 20 yards by 40 yards. What is its area?

$$A = 20 \times 40 = 800 yd$^2$$

Q: A box has dimensions 23 cm by 18 cm by 4 cm. What is its volume?

$$V = 23 \times 18 \times 4 = 1656 cm$^3$$

Q: A parallelogram has a base of 200 ft and a height of 120 ft. What is its area?

$$A = 200 \times 120 = 24,000 ft$^2$$

Q: A city park is a rectangle 55 yards by 39 yards. How much sod is needed to cover the park?

$$A = 55 \times 39 = 2,145 yd$^2$$

Question 1

Suppose you want to measure the perimeter of a fern leaf, which has jagged edges. If you use a ruler with smaller and smaller units, what will you observe about your measurement?

Accepted Answer

The measured perimeter will increase as the ruler units get smaller.

Question 2

Which formula correctly gives the area of a triangle with base $$b$$ and height $$h$$?

Accepted Answer

$$A = \frac{1}{2} b h$$

Question 3

A window consists of a rectangle 4 ft wide and 6 ft tall, capped by a semicircle. What is the total perimeter of the window (to the nearest foot)? Use $$\pi \approx 3.14.$$

Accepted Answer

Perimeter $$= 2 	imes 6 + 4 + (\pi 	imes 2) = 18 + 6.28 = 24.28 ft$$

Question 4

What is the area of a parallelogram with base $$b = 6 m $$and height $$h = 3 m$$?

Accepted Answer

$$A = 6 	imes 3 = 18 m$^2$$

Question 5

A triangle has sides of 5 cm, 5 cm, and 6 cm, with a height of 4 cm to the base of 6 cm. What is its area?

Accepted Answer

$$A = \frac{1}{2} 	imes 6 	imes 4 = 12 cm$^2$$

Question 6

What is the sum of the interior angles of any triangle?

Accepted Answer

$$180^\circ$$

Question 7

A rectangular garden is 20 yards by 40 yards. What is its area?

Accepted Answer

$$A = 20 	imes 40 = 800 yd$^2$$

Question 8

A box has dimensions 23 cm by 18 cm by 4 cm. What is its volume?

Accepted Answer

$$V = 23 	imes 18 	imes 4 = 1656 cm$^3$$

Question 9

If you double the height, width, and depth of a cube, by what factor does its volume increase?

Accepted Answer

By a factor of 8

Question 10

A right triangle has legs of 12 ft and 9 ft. What is the length of the hypotenuse?

Accepted Answer

$$c = \sqrt{12^2$ + 9^2$} = \sqrt{225} = 15 ft$$

Question 11

Which of the following best describes the relationship between the surface area and volume of similar objects as their size increases?

Accepted Answer

Volume increases faster than surface area.

Question 12

A parallelogram has a base of 200 ft and a height of 120 ft. What is its area?

Accepted Answer

$$A = 200 	imes 120 = 24,000 ft$^2$$

Question 13

A right circular cylinder has a radius $$r$$ and height $$h. $$What is the formula for its volume?

Accepted Answer

$$V = \pi r^2$ h$$

Question 14

A triangle has sides of 9, 12, and unknown $$a. A $$similar triangle has sides 4, 6, and $$c'. $$What is the value of $$a$$?

Accepted Answer

$$a = 8$$

Question 15

A city park is a rectangle 55 yards by 39 yards. How much sod is needed to cover the park?

Accepted Answer

$$A = 55 	imes 39 = 2,145 yd$^2$$

Question 16

If a child grows 30% bigger in all dimensions, by what factor does his weight increase? (Assume weight scales with volume.)

Accepted Answer

By a factor of $$2.197$$

Question 17

A road rises 2 ft for every 20 ft of horizontal distance. What is the slope and grade of the road?

Accepted Answer

Slope $$= 0.1$$, Grade $$= 10\%$$

Question 18

Which shape gives the maximum area for a fixed perimeter?

Accepted Answer

A circle

Modeling with Geometry: Quantitative Reasoning Study Notes

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