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HSPT Math : How to find the area of a figure

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Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.?

Possible Answers:

1936 – 484π dollars

7744 – 1936π dollars

1936 dollars

5808 dollars

1936π dollars

Correct answer:

1936 – 484π dollars

Explanation:

The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in².

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr² or π * 44² = 1936π in². Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars.

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Example Question #311 : Plane Geometry

A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.20 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 25 in.?

Possible Answers:

500 - 125π dollars

625 - 25π dollars

625 dollars

2500 - 625π dollars

500 dollars

Correct answer:

500 - 125π dollars

Explanation:

The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 50 inches for our problem. Its total area would be 50 * 50 or 2500 in².

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr² or π * 25² = 625π in². Therefore, the area remaining would be 2500 - 625π. The cost of the waste would be 0.2 * (2500 – 625π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 5 from our subtraction. This would give us: 0.2 * 5 * (500 – 125π). Since 0.2 is equal to 1/5, 0.2 * 625 = 125. Therefore, our final answer is: 500 – 125π dollars.

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Example Question #551 : Geometry

Mary has a decorative plate with a diameter of ten inches. She places the plate on a rectangular placemat with a length of 18 inches and a width of 12 inches. How much of the placemat is visible?

Possible Answers:

$216-25\pi\hspace{1 mm}inches^2$

$191\pi\hspace{1 mm}inches^2$

$216\pi\hspace{1 mm}inches^2$

$25\pi\hspace{1 mm}inches^2$

$216\hspace{1 mm}inches^2$

Correct answer:

$216-25\pi\hspace{1 mm}inches^2$

Explanation:

First we will calculate the total area of the placemat:

$A=l\times w= 18\times 12= 216\hspace{1 mm}inches^2$

Next we will calculate the area of the circular place

$A=\pi r^2$

And

$d=2r=10$

$r=5\hspace{1 mm}inches$

$A=\pi r^2=\pi (5^2)=25\pi\hspace{1 mm}inches^2$

We will subtract the area of the plate from the total area

$216-25\pi\hspace{1 mm}inches^2$

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Example Question #21 : How To Find The Area Of A Circle

Allen was running around the park when he lost his keys. He was running around aimlessly for the past 30 minutes. When he checked 10 minutes ago, he still had his keys. Allen guesses that he has been running at about 3m/s.

If Allen can check 1 square kilometer per hour, what is the longest it will take him to find his keys?

Possible Answers:

$\sim 1,000\ hours$

$\sim 81\ hours$

$\sim 10\ hours$

$\sim 3\ hours$

$\sim 100\ hours$

Correct answer:

$\sim 10\ hours$

Explanation:

Allen has been running for 10 minutes since he lost his keys at 3m/s. This gives us a maximum distance of 3*600=1800m from his current location. If we move 1800m in all directions, this gives us a circle with radius of 1800m. The area of this circle is

$1800^2\pi \approx 10,178,760m^2$

Our answer, however, is asked for in kilometers. 1800m=1.8km, so our actual area will be $\frac{10,178,760}{1,000,000} = 10.2$ square kilometers. Since he can search 1 per hour, it will take him at most 10.2 hours to find his keys.

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Example Question #11 : How To Find The Area Of A Figure

Find the area of a square if the length is 2-x .

Possible Answers:

4-x^2

x^2+4x-4

8-4x

x^2-4x+4

4-2x

Correct answer:

x^2-4x+4

Explanation:

The area of a square is:

A=s^2

Substitute the length and simplify.

A=(2-x)^2 = (2-x)(2-x)

A=(2)(2)+(2)(-x)+(-x)(2)+(-x)(-x)

A=4-2x-2x+x^2 = x^2-4x+4

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Example Question #12 : Geometry

A square has area 64. Which of the following gives 25% of the length of one of its diagonals?

Possible Answers:

$2\sqrt{2}$

$4\sqrt{2}$

$8\sqrt{2}$

$\sqrt{2}$

Correct answer:

$2\sqrt{2}$

Explanation:

A square with area 64 has as the length of one of its sides the square root of this area, which is $\sqrt{64} = 8$ . The length of a diagonal of a square is $\sqrt{2}$ times this sidelength, so the square has diagonals of length $8 \sqrt{2}$ .

25% of this is

$8 \sqrt{2} \cdot \frac{25}{100} = \frac{8 \sqrt{2}}{1} \cdot \frac{1}{4} = \frac{8\sqrt{2}}{4} = 2 \sqrt{2}$ .

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Example Question #221 : Problem Solving

Your backyard is 6ft wide and 10ft long, what is its area?

Possible Answers:

$16ft^{2}$

$50ft^{2}$

$10ft^{2}$

$60ft^{2}$

Correct answer:

$60ft^{2}$

Explanation:

The area of a rectangle is

Area=length*width .

So for this you just multiple those two values together to get,

L=10, w=6

$A=L\cdot w$

6*10=60 .

Remeber that the units of area are squared.

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Example Question #221 : Hspt Mathematics

If a trianlge has a base of and a height of , the area is . True or false?

Possible Answers:

False

True

Correct answer:

False

Explanation:

The area of a triangle is

$A=\frac{1}{2}*base*height$ .

is equal to the base multiplied by the height.

So you still need to multiple by $\frac{1}{2}$ or divide by .

This gives you an actual area of .

Therefore the statement is false.

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Example Question #11 : How To Find The Area Of A Figure

Target

A blindfolded man throws a dart at the above target. Assuming the dart hits the target, and that no skill is involved, give the probability that the dart will land inside a blue region.

Possible Answers:

$\frac{4}{9}$

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{2}{9}$

Correct answer:

$\frac{1}{4}$

Explanation:

The reasoning for the answer can best be seen by adding two more lines as shown in purple in the diagram below:

Target

The target is now divided into sixteen squares of equal size, four of which are blue. This makes the probability that the dart will land in the blue region

$\frac{4}{16} = \frac{1}{4}$ .

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Example Question #221 : Problem Solving

Spinner target

Above is a target. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A blindfolded man throws a dart at the above target. Assuming the dart hits the target, and that no skill is involved, give the odds against the dart landing in the yellow region.

Possible Answers:

14 to 1

39 to 1

13 to 1

40 to 1

Correct answer:

39 to 1

Explanation:

Call the radius of one of the smaller quarter-circles 1 (the reasoning is independent of the actual radius). Then the area of each quarter-circle is

$\frac{1}{4} \cdot \pi r ^{2} = \frac{1}{4} \cdot \pi \cdot 1 ^{2} = \frac{1}{4} \pi$ .

Each of the four wedges of one such quarter-circle has area

$\frac{1}{4} \cdot \frac{1}{4} \pi = \frac{\pi}{16}$ .

The yellow region is one such wedge.

The radius of each of the larger quarter-circles is 2, so the area of each is $\frac{1}{4} \cdot \pi r ^{2} = \frac{1}{4} \cdot \pi \cdot 2 ^{2} = \pi$

The total area of the target is

$\frac{1}{4}\pi + \frac{1}{4} \pi + \pi + \pi = \frac{5 \pi}{2}$

Therefore, the yellow wedge is

$\frac{\pi}{16} \div \frac{5 \pi}{2} = \frac{\pi}{16} \cdot \frac{2}{5 \pi}= \frac{1}{8} \cdot \frac{1}{5 }= \frac{1}{40}$

of the target, and the odds against the dart landing in that region are 39 to 1.

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HSPT Math : How to find the area of a figure

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

Example Question #311 : Plane Geometry

Example Question #551 : Geometry

Example Question #21 : How To Find The Area Of A Circle

Example Question #11 : How To Find The Area Of A Figure

Example Question #12 : Geometry

Example Question #221 : Problem Solving

Example Question #221 : Hspt Mathematics

Example Question #11 : How To Find The Area Of A Figure

Example Question #221 : Problem Solving

All HSPT Math Resources

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