Can someone help me with these two algebra word problems?

Question

1. The area of a circle is given by the formula A=[pi]r^2, where r is the radius. If increasing the radius of a circle by 1 inch gives the resulting circle an area of 100pi sq in., what is the radius of the original circle?

2. Mikaela placed a frame around a print that measures 10 in by 10 in. The area of just the frame itself is 69 sq in. What is the width of the frame?

Answer 1

1. r is the original radius. r + 1 is the new radius.

A = π r²
100π = π (r + 1)² = π (r² + 2r + 1)
100 = r² + 2r + 1
0 = r² + 2r - 99
0 = (r + 11)(r - 9)

Throw out the -11, so r = 9.

(This could also be done very easily be saying, what value of r produces an area of 100π? 10 of course...so the original radius must be 10 - 1 = 9.)

2. If the "extra" of the frame is x (1/2 of x on one side, 1/2 on the other), then both the length and the width of the frame are 10 + x. Then, to find the area of just the frame, take the area of the frame and the print (10 + x)² and subtract out the area of the print (100), and set it equal to 69:

(10 + x)² - 100 = 69
x² + 20x + 100 - 100 = 69
x² + 20x - 69 = 0
(x + 23)(x - 3) = 0

Throw out the -23, so x = 3, and the width of the frame is 10 + 3 = 13. (Which means that the frame is 1.5 inches bigger than the print on all sides.)

(And yes, as everyone else pointed out, this can be done more easily by adding the area of the picture to the area of the frame to get 169, so each side is 13, so the frame is (13 - 10)/2 = 1.5 inches bigger than the picture.)

Answer 2

1. There are two variables in this equation, area and radius.
First circle: area = A, radius = r. these are just arbitrary names, you can call the variables anything you want.
Second circle: area = A+100, radius = r+1
see how I changed the variables from the first equation?
Now we end up with two equations:

A = (pi)r^2
A+100 = (pi)(r+1)^2

These represent the first and second circle, respectively. Two equations, two variables. That means you have enough information to solve for r, which will give you your answer (hint: subtract 100 from both sides of equation 2, then use the resulting (pi)r^2 = (pi)(r+1)^2 -100).

2. This is a bad question. You have to assume that the frame is going to be square, since the picture is square, which is not always going to be true. Let's assume that it's true to make it simple, though.

Let's make a mental image. Draw a square on a piece of paper for me. Now using the same center, draw a bigger square around it. This is the problem we're working with. The area of the small square (the picture) is 100 (10x10). 69 represents the area that is in the big square but is NOT in the small square (the part between the lines). If we add 100 to 69, we find the area of the big square, which is 169. Assuming that the big square is, in fact, a square, then the width (which is the same length as the height!) has to be equal to the square root of 169, which is 13.

Answer 3

Let A be the original area
1.
Let r be the original radius
Let r+1 be the new radius, then
new area = pi(r+1)^2 = A (old area) + 100
but old area A = pi(r^2), so
pi(r+1)^2 = pi(r^2) + 100
pi(r^2+2r+1) = pi(r^2) + 2pi(r) + pi = pi(r^2) + 100
2pi(r) = 100 - pi
r = 50/pi - 1/2

2.
Let x = width of the frame from the print edge to the outer perimeter of the frame
The frame minus the print is four rectangles whose areas are:
top: (10 + 2x)x
right side: (10)x
left side: (10)x
bottom: (10 + 2x)x
The total area is 4x^2+40x = 69; solving for x using the quadratic formula yields
x = {-10 +- 2sqr[94]}/2 = -5 +- sqr[94]
The rest is easy

Answer 4

1. Since Πr^2 is the area, and your area is 100Π, you can write it as Πr^2 = 100Π. Now, divide both sides by Π, and you get r^2 = 100. Take the square of both sides, and you get r = +- 10, which you know is 10, because a radius can't be negative. So, the radius of the curcle after you added an inch is 10. That minus 1 is 9 =)

2. Assuming the 10x10 picture fits perfectly into the frame, which I'm assuming is a square with a 10x10 inch square in the middle of it, you get 169 as the total area. The square root of 169 is 13, which would be the width of the frame, I suppose. If that's not what you wanted, that also means the frame's border is 1.5 inches.

Answer 5

Answer 1:
Let original radius = r
Let the new radius be 'r + 1'
pi*(r + 1)^2 = 100 pi
(r + 1)^2 = 100 pi/pi
(r + 1)^2 = 100
r^2 + 2r + 1 = 100
r^2 + 2r - 99 = 0
r^2 + 11r - 9r - 99 = 0
r(r + 11) - 9(r + 11) = 0
(r + 11)(r - 9) = 0
Either r + 11 = 0
or r - 9 = 0

If r + 11 = 0
r = -11
This isn't possible as radius can never be negative

If r - 9 = 0
r = 9
The original radius is 9 inches.

Answer 6

1. Easy one
Pi(r+1)squared = 100Pi
divide both sides by pi

(r+1)squared = 100
taking the square root of both sides
r+1 = +/-10
since no measurement is negative we use +10
r = 10-1
r=9 inches

2. Harder
area of picture = 10*10 = 100
area of picture frame = 4*(10*width)+4*(w*w)=69
divide both side by 4.. we have
(w)squared+10w-(69/4)=0
and we have a quadratic equation
use quadratic formula
-b + square root[(b)squared - 4ac] / 2a
let a = 1 b=10 c=-69/4
-10 + sr(100+69)/2
[-10 + sr(169)]/2 = 3/2
w is equal to 1.5 inches

Answer 7

1. r is the original radius. r + 1 is the new radius.
We know that
A = pi r^2
100 = pi (r + 1)^2 = pi (r^2 + 2r + 1)
100/pi = r^2 + 2r + 1
0 = r^2 + 2r + (1 - 100/pi)

Use quadratic equation roots formula we get..............

r = (-2 +/- sqrt(4 - 4(1 - 100/pi)))/2
r = (-2 +/- 11.284)/2
r = 4.642

2. If the "width" of the frame is x, then both the length and the width of the frame are 10 + 2x,because the width adds from both sides of length and breadth, then you subtract out the area of the print:

(10+2x)^2-10*10=69
(10+2x)^2=169=13^2
thus by taking positive roots,we get......

10+2x=13
=>2x=3
=>x=1.5 in

Thus width of frame=1.5inch

Answer 8

let A = 100

100 = (pi)(r + 1)^2
-1 = -1
99=(pi)r^2
/pi=/pi

99/pi= sqr(r^2)

first off the frame is too small since the picture is 100 sq in and assuming that the frame is a prefect square it would be sqr(69) = 8.3 and some change.

Answer 9

Area = pi x radius squared

100 = pi x radius squared

radius squared = 100/3.1416

radius squared = 31.8

radius = root of 31.8 = 5.64

5.64 - 1 = 4.64 (answer)

Answer 10

one inch more fadius gives
100pi=[pi](r+1)^2 so solve this equaltion for r and youir are done