Answer:
1. A ring of grass with an area of 314 yd squared surrounds a circular flower bed. Find the width x of the ring of grass.
The radius of the circular flower bed = 10 yd.
2. Harriet has 80 m of fencing materials to enclose three sides of a rectangular garden. She will use the side of her garage as a border for the fourth side. Find the width x of the garden if its area is to be 700 m squared.
3. Sid cuts four congruent squares from the corners of a 30-in.-by-50-in. rectangular piece of cardboard so that it can be folded to make a box. Find the side length s of the squares, given that teh area of the bottom of the box is 200 in squared.
Step-by-step explanation:
We want to get the width of the ring of grass, we will get x = 4.15 yd.
Some information is missing here, we know that the radius of the flower bed is 10yd.
Remember that the area of a circle of radius R is given by:
A = 3.14*R^2
So let's say that the circular flower bed has a radius R and if we also include the ring of grass, we have a radius R'.
Then the area of the ring of grass is given by:
A = 3.14*(R'^2 - R^2) = 314yd^2
Notice that R' - R would be the width of the ring of grass.
We can rewrite:
R'^2 - R^2 = (R' - R)*(R' + R) = x*(R' - R + 2*R) = x*(x + 2*R)
Replacing that in the equation of the area, we get:
3.14*(x*(x + 2*R)) = 314yd^2
(x*(x + 2*R)) = (314yd^2)/3.14 = 100yd^2
x*(x + 2*R) = 100 yd^2
Then we have:
x^2 + x*2*R = 100 yd^2
And remember that the radius of the flower bed is 10yd, then we have:
x^2 + x*2*10yd = 100 yd^2
x^2 + (20yd)*x - 100 yd^2 = 0
This is just a quadratic equation, the solutions are given by:
[tex]x = \frac{-20yd \pm \sqrt{(20yd)^2 - 4*1*(-100yd^2)} }{2*1} \\\\x = \frac{-20yd \pm 28.3yd }{2}[/tex]
We only take the positive solution, which is:
x = (-20yd + 28.3yd)/2 = 4.15yd
The width of the flower ring is equal to 4.15 yd
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