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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

A circular pool measures 10 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 3 inches, how wide will the border be?

I know that 1 cubic yard is 27 cubic feet.

I say let x = width of the border around the pool

I don't know where to go with this information.

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**amnkb****Member**- Registered: 2023-09-19
- Posts: 175

harpazo1965 wrote:

A circular pool measures 10 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 3 inches, how wide will the border be?

I know that 1 cubic yard is 27 cubic feet.

I say let x = width of the border around the pool

I don't know where to go with this information.

what is the formula for the area of a circle with radius r?

if x is the border width then what expression gives the outer radius of the border?

how much of a foot is 3 inches?

the area of the border is (outer circle - inner circle)

the volume is area times thickness

plug info into equation, set equal to 27

solve for x

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

amnkb wrote:

harpazo1965 wrote:A circular pool measures 10 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 3 inches, how wide will the border be?

I know that 1 cubic yard is 27 cubic feet.

I say let x = width of the border around the pool

I don't know where to go with this information.what is the formula for the area of a circle with radius r?

if x is the border width then what expression gives the outer radius of the border?

how much of a foot is 3 inches?

the area of the border is (outer circle - inner circle)

the volume is area times thickness

plug info into equation, set equal to 27

solve for x

Outline:

1. A = pi•r^2

2. The expression that gives the outer radius is not too clear to me.

I think it is: A = pi•(2x + 10)^2

3. A foot is 12 inches. How much of a foot is 3 inches? It is 3/12 of a foot.

4. Let A_b = area of border. I think it is pi(2x + 10)^2 - pi•(5)^2

5. Is the right equation the following?

27 = pi(2x + 10)^2 - pi•(5)^2, where x is the width of the border.

Yes? If not, can you please set up the correct equation? I can then take it from there.

Thank you.

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**amnkb****Member**- Registered: 2023-09-19
- Posts: 175

harpazo1965 wrote:

Outline: 1. [The area A of a circle with radius r is given by] A = pi•r^2

yes

harpazo1965 wrote:

2. The expression that gives the outer radius is not too clear to me.

I think it is: A = pi•(2x + 10)^2

draw the circle for the pool

label the radius

draw the outer cricle for the border

label the width

label the radius

is the radius of the pool+border area equal to the whole diameter of the pool plus the border width on both sides?

harpazo1965 wrote:

3. A foot is 12 inches. How much of a foot is 3 inches? It is 3/12 of a foot.

3/12 = 1/4 = 0.25

harpazo1965 wrote:

4. Let A_b = area of border. I think it is pi(2x + 10)^2 - pi•(5)^2

no

harpazo1965 wrote:

5. Is the right equation the following?

27 = pi(2x + 10)^2 - pi•(5)^2, where x is the width of the border.

where does this use the thickness of the border?

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

amnkb wrote:

harpazo1965 wrote:Outline: 1. [The area A of a circle with radius r is given by] A = pi•r^2

yes

harpazo1965 wrote:2. The expression that gives the outer radius is not too clear to me.

I think it is: A = pi•(2x + 10)^2draw the circle for the pool

label the radius

draw the outer cricle for the border

label the width

label the radius

is the radius of the pool+border area equal to the whole diameter of the pool plus the border width on both sides?harpazo1965 wrote:3. A foot is 12 inches. How much of a foot is 3 inches? It is 3/12 of a foot.

3/12 = 1/4 = 0.25

harpazo1965 wrote:4. Let A_b = area of border. I think it is pi(2x + 10)^2 - pi•(5)^2

no

harpazo1965 wrote:5. Is the right equation the following?

27 = pi(2x + 10)^2 - pi•(5)^2, where x is the width of the border.

where does this use the thickness of the border?

Radius of pool = 5 feet

Border area = pi(x + 5)^2

Radius of pool + border area = 2x + 10 feet.

So, 5 + pi(x + 5)^2 = 2x + 10

Correct equation?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,863

hi

This problem is made more complicated because it switches from yards, to feet to inches. To get a correct equation everything needs to be in the same units. So I'll convert to feet.

Radius of pool = 5 ft

Let the width of the border (what we want to find) be x ft. Then radius out to the outer edge is 5 + x

Area of the circle out to the outer edge is

So area of the concrete ring is

The concrete is 3 inch thick = 1/4 ft. The volume of concrete is 1 cu yard = 3 x 3 x 3 = 27 cu ft.

Volume of the border concrete = area x thickness hence

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

Bob wrote:

hi

This problem is made more complicated because it switches from yards, to feet to inches. To get a correct equation everything needs to be in the same units. So I'll convert to feet.

Radius of pool = 5 ft

Let the width of the border (what we want to find) be x ft. Then radius out to the outer edge is 5 + xArea of the circle out to the outer edge is

So area of the concrete ring is

The concrete is 3 inch thick = 1/4 ft. The volume of concrete is 1 cu yard = 3 x 3 x 3 = 27 cu ft.

Volume of the border concrete = area x thickness hence

Bob

Wow! Wish I could break down the problem like you just did. I did get parts of it

correct but guessing, of course. This has been my problem with mathematics for years. I know to let x be what we are looking for but forming the needed equation(s) to solve applications has been a struggle for me. I can take it from here. Thanks for the equation.

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**forbuytoo****Novice**- Registered: 2023-11-11
- Posts: 4

Please explain the final figure 27.

Is this figure meant to be inches or feet

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,863

hi forbuytoo

Welcome to the forum.

I'm converting all the measurements into feet. The concrete delivery is 1 cubic yard. There are 3 feet in a yard, so a cubic yard is 3 feet by 3 feet by 3 feet = 27 cubic feet. Think of that 'cube' as like a rubic cube, 3 layers of cubic feet each measuring 3 by 3. The answer for x will be in feet too, so it might need converting back to inches for a useful answer.

I once ordered some readymix concrete to make a base for a garden shed. I measured up in feet and worked out that one cubic yard would just do the job. On the 'phone, the man said "Our minimum order is one cubic metre".

So I ordered that. Then I did the sums. Even though a yard is just short of a metre, by the time you've 'cubed' it, the excess concrete that I'd just ordered was quite a lot. EeeK! Shows one should always do the sums first. So I decided to widen the base to use up the excess. Ok no problem. Except it was when it arrived. In my head I had imagined a neat pyramid of slightly runny concrete, which I could quickly spread out across the area. Not so. Readymix is really runny ... I suppose it has to be or it might set inside the lorry's drum. So this huge 'cow pat' just spread out and out and out. Luckily my sons were around so we all got scooping and spreading and saved it from going all over the garden. I pass on this experience so that others can avoid my mistakes.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**forbuytoo****Novice**- Registered: 2023-11-11
- Posts: 4

The border is 3 inches deep but I still cannot see how wide will it be.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,863

In post #6 I have derived an equation for working this out. Do you understand it?

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

Bob wrote:

In post #6 I have derived an equation for working this out. Do you understand it?

Bob

Yes, Bob I understand how to work it out. I will show my work on days off. I work 32 weekend hours (two back to back double shifts). I am busy and seriously tired.

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**forbuytoo****Novice**- Registered: 2023-11-11
- Posts: 4

I think Bob's last post was meant for me. I am not a mathematician and admit I am not following the formula too well and was simply interested in finding the answer to the problem set by sologuitar. i.e. what is the width of the border? Thanks

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

forbuytoo wrote:

I think Bob's last post was meant for me. I am not a mathematician and admit I am not following the formula too well and was simply interested in finding the answer to the problem set by sologuitar. i.e. what is the width of the border? Thanks

Thanks for letting me know.

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**forbuytoo****Novice**- Registered: 2023-11-11
- Posts: 4

looking forward to seeing the answer.

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

forbuytoo wrote:

looking forward to seeing the answer.

Bob provided the needed equation to find x, which represents the width of the border. Used Wolfram to find x.

Here it is again:

pi((5 + x)^2 - 5^2)(1/4) = 27

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**amnkb****Member**- Registered: 2023-09-19
- Posts: 175

sologuitar wrote:

forbuytoo wrote:

looking forward to seeing the answer.

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

amnkb wrote:

sologuitar wrote:forbuytoo wrote:looking forward to seeing the answer.

Wow! Nicely-done! I like your step by step process to reach the answer.

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