Math / Geometry

QuestionGoogle Account ALEKS - Carlos Silva-Villa - Learn
Area and Volume Word problem involving the area between two concentric circles
A flower garden is shaped like a circle. Its radius is 16 yd . A ring-shaped path goes around the garden. The width of the path is 5 yd.
The gardener is going to cover the path with sand. If one bag of sand can cover 5 yd ${ }^{2}$ , how many bags of sand does the gardener need? Note that sand comes only by the bag, so the number of bags must be a whole number. (Use the value 3.14 for $\pi$ .) $\square$ bags of sand

Studdy Solution

STEP 1

What is this asking? How many bags of sand do we need to cover a circular path, given its width and the radius of the inner circle? Watch out! Don't confuse the radius of the inner circle with the radius of the outer circle!

STEP 2

1. Find the area of the big circle
2. Find the area of the small circle
3. Find the area of the path
4. Calculate the number of sandbags

STEP 3

The radius of the small circle is $\text{16 yd}$ .
The path around it is $\text{5 yd}$ wide.
So, the radius of the big circle is $\text{16 yd} + \text{5 yd} = \text{21 yd}$ .
We add the width of the path to the small circle's radius to get the big circle's radius!

STEP 4

The area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
The radius of the big circle is $\text{21 yd}$ .
Using $\pi \approx 3.14$ , the area is $3.14 \cdot (21 \text{ yd})^2 = 3.14 \cdot 441 \text{ yd}^2 = 1384.74 \text{ yd}^2$ .
We've got the area of the big circle!

STEP 5

The radius of the small circle is $\text{16 yd}$ .
Using $\pi \approx 3.14$ , the area of the small circle is $3.14 \cdot (16 \text{ yd})^2 = 3.14 \cdot 256 \text{ yd}^2 = 803.84 \text{ yd}^2$ .
Awesome!

STEP 6

The area of the path is the difference between the area of the big circle and the area of the small circle.
That's $1384.74 \text{ yd}^2 - 803.84 \text{ yd}^2 = 580.9 \text{ yd}^2$ .
We're getting closer to figuring out how much sand we need!

STEP 7

Each bag of sand covers $5 \text{ yd}^2$ .
We need to cover $580.9 \text{ yd}^2$ .
So, we divide the total area by the area covered by one bag: $\frac{580.9 \text{ yd}^2}{5 \text{ yd}^2/\text{bag}} = 116.18 \text{ bags}$ .

STEP 8

Since we can only buy whole bags of sand, we need to round up to the nearest whole number.
So, we need 117 bags of sand.
We did it!

STEP 9

The gardener needs 117 bags of sand.

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Studdy Solution

STEP 1

What is this asking? How many bags of sand do we need to cover a circular path, given its width and the radius of the inner circle? Watch out! Don't confuse the radius of the inner circle with the radius of the outer circle!

STEP 2

1. Find the area of the big circle
2. Find the area of the small circle
3. Find the area of the path
4. Calculate the number of sandbags

STEP 3

The radius of the small circle is $\text{16 yd}$ .
The path around it is $\text{5 yd}$ wide.
So, the radius of the big circle is $\text{16 yd} + \text{5 yd} = \text{21 yd}$ .
We add the width of the path to the small circle's radius to get the big circle's radius!

STEP 4

The area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
The radius of the big circle is $\text{21 yd}$ .
Using $\pi \approx 3.14$ , the area is $3.14 \cdot (21 \text{ yd})^2 = 3.14 \cdot 441 \text{ yd}^2 = 1384.74 \text{ yd}^2$ .
We've got the area of the big circle!

STEP 5

The radius of the small circle is $\text{16 yd}$ .
Using $\pi \approx 3.14$ , the area of the small circle is $3.14 \cdot (16 \text{ yd})^2 = 3.14 \cdot 256 \text{ yd}^2 = 803.84 \text{ yd}^2$ .
Awesome!

STEP 6

The area of the path is the difference between the area of the big circle and the area of the small circle.
That's $1384.74 \text{ yd}^2 - 803.84 \text{ yd}^2 = 580.9 \text{ yd}^2$ .
We're getting closer to figuring out how much sand we need!

STEP 7

Each bag of sand covers $5 \text{ yd}^2$ .
We need to cover $580.9 \text{ yd}^2$ .
So, we divide the total area by the area covered by one bag: $\frac{580.9 \text{ yd}^2}{5 \text{ yd}^2/\text{bag}} = 116.18 \text{ bags}$ .

STEP 8

Since we can only buy whole bags of sand, we need to round up to the nearest whole number.
So, we need 117 bags of sand.
We did it!

STEP 9

The gardener needs 117 bags of sand.

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Studdy Solution

STEP 1

What is this asking? How many bags of sand do we need to cover a circular path, given its width and the radius of the inner circle? Watch out! Don't confuse the radius of the inner circle with the radius of the outer circle!

STEP 2

1. Find the area of the big circle2. Find the area of the small circle3. Find the area of the path4. Calculate the number of sandbags

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

Since we can only buy whole bags of sand, we need to round **up** to the nearest whole number.So, we need **117** bags of sand.We did it!

STEP 9

The gardener needs 117 bags of sand.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Find the area of the big circle
2. Find the area of the small circle
3. Find the area of the path
4. Calculate the number of sandbags

Since we can only buy whole bags of sand, we need to round up to the nearest whole number.
So, we need 117 bags of sand.
We did it!