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Question 447437: Not sure exactly which category this goes under. I have a very long math problem I'm struggling to solve. I've googled and seen parts of it or similar problems, but I haven't seen the full thing, so I'm not entirely sure how to go about solving this. Any help will be appreciated, so I thank you in advance. I will also post the parts I think I already worked out so that you can see what I've done so far.
A gardener has a rose garden that measures 30 feet by 20 feet. He wants to put a uniform border of pine bark around the outside of the garden. Let x be the width of the border in feet. Write an expression for the total area of the border in terms of the width x.
I came up with 4x^2 + 100x as my answer. Correct?
Find out how wide the border should be if he has enough pine bark to cover 336 square feet.
I came up with 3 feet as my answer. Correct?
If the border needs to be covered to a depth of 3 inches and pine bark costs $29 per cubic yard, what will the material cost to lay the pine bark border? Assume that you must purchase a whole number of cubic yards and not a fraction of a cubic yard.
Here's where I'm struggling with *how* to answer this one. I came up with $6786, but I'm thinking that's way off. Any advice or suggestions?
Also, now assume you can actually buy as much or as little bark as you need to cover the border area, that is, the vendor will sell even a small fraction of a cubic yard if you want. Write an expression for the cost in dollars as a function of the width x, of the border in feet. Assume again that the border is to be covered to a depth of 3 inches and that pine bark costs $29 per cubic yard.
Please help! :) Thanks!
(There are actually two more questions to this problem, but I'm hoping the above, when I'm able to figure it out, will help me in solving them.)
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! A gardener has a rose garden that measures 30 feet by 20 feet.
He wants to put a uniform border of pine bark around the outside of the garden. Let x be the width of the border in feet.
Write an expression for the total area of the border in terms of the width x.
I came up with 4x^2 + 100x as my answer. Correct?
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Draw the picture:
You have a rectangle inside a rectangle.
The area of the larger rectangle is (30+2x)(20+2x)
The area of the inside rectangle is 30*20
So the area of the border is (30+2x)(20+2x)-30*20 = 4x^2+100x sq. ft.
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Find out how wide the border should be if he has enough pine bark to cover 336 square feet.
I came up with 3 feet as my answer. Correct?
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Solve 4x^2+100x = 336
x^2+25x-84 = 0
(x-3)(x+28) = 0
Positive solution:
x = 3 ft.
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If the border needs to be covered to a depth of 3 inches and pine bark costs $29 per cubic yard, what will the material cost to lay the pine bark border? Assume that you must purchase a whole number of cubic yards and not a fraction of a cubic yard.
Volume = (area)(depth) = 336 sq.ft * (0.25 ft) = 84 cu. ft
84 cu ft = 84/27 = 3.111 cu yrd
Price = 3.1111*$29 = $90.22
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Also, now assume you can actually buy as much or as little bark as you need to cover the border area, that is, the vendor will sell even a small fraction of a cubic yard if you want. Write an expression for the cost in dollars as a function of the width x, of the border in feet. Assume again that the border is to be covered to a depth of 3 inches and that pine bark costs $29 per cubic yard.
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Volume = (area)(depth)
Volume = [(4x^2+100x)(1/4)]/27 = (27/4)(4x^2+100x) cu. yds
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Cost = (29*27/4)(4x^2-100x) = 195.75*4(x^2-25x) = 783(x^2-25x) dollars
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Cheers,
Stan H.
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