1) A corner lot that originally was square lost 185m^2 of area when one of the adjacent streets was widened by 3 m and the other was widened by 5 m. Find the new dimensions of the lot. (Hint: Let x = the lenght of a side of the original square lot. )
2) A running track 4 m wide goes around a soccer field that is twice as long as it is wide. At each end of the soccer field the track is a semi-circle with inner radius r. Find a formula for the area of the track in terms of *pi and r.
3a) Suppose that you plan to run once around the track described in the problem above. If you stay 0.5 m from the inner edge of the track, how far will you run? (Hint: The circumference of a circle is *2pir. Your answer will be in terms of *pi and r.)
3b) Suppose that a friend stays 0.5 m from the outer edge of the track. How much farther does your friend run than you do?
*pi = 3.14 or 22/7
original square ---- > let each side be x
new sides are x-3 and x-5
so x^2 - (x-3)(x-5) = 185
x^2 - (x^2 - 8x + 15) = 185
8x - 15 = 185
x = 200/8 = 25
check:
original square was 25 by 25 , area = 625
new rectangle was 22 by 20 , area = 440
difference = 625-440 = 185
All looks good
2. I will let you do this one.
Hint: let the width of the soccer field be 2r, that way you can label the radius of the ends as r
The two semi-circular ends will constitute one whole circle. So the circular part of the track will be the "ring" formed by two circles, one of radius r, the other of radius (r-4)
1) To find the new dimensions of the lot, we can use the information given in the problem. Let's assume that the original side length of the square lot is x.
When one adjacent street is widened by 3 m, the new length of the lot becomes (x + 3). Similarly, when the other adjacent street is widened by 5 m, the new width of the lot becomes (x + 5).
We know that the original area of the square lot is x^2, and the new area of the lot after the streets are widened is (x + 3)(x + 5).
According to the problem, the new area is 185m^2 less than the original area. So, we can set up the following equation:
x^2 - 185 = (x + 3)(x + 5)
To solve this equation, we can distribute and simplify:
x^2 - 185 = x^2 + 8x + 15
Move all the terms to one side:
0 = 8x + 15 - 185
Simplify further:
8x = 170
x = 170/8
Therefore, the original side length of the square lot is approximately 21.25 m.
Substituting this value into the dimensions of the new lot:
Length = x + 3 = 21.25 + 3 = 24.25 m
Width = x + 5 = 21.25 + 5 = 26.25 m
So, the new dimensions of the lot are approximately 24.25 m by 26.25 m.
2) To find the formula for the area of the track, we need to understand the shape and dimensions of the track.
The track consists of two straight sections (width = 4 m) along the length of the soccer field and two semi-circular sections at each end. The soccer field is twice as long as it is wide. Let's assume the width of the soccer field is w.
The length of the soccer field is then 2w.
Now, let's look at one of the semi-circular sections at the end of the soccer field. The inner radius of the semi-circle is given as r. The width of the track is 4 m, so the outer radius of the semi-circle is (r + 4).
The area of the semi-circle can be calculated using the formula:
Area = (1/2) * pi * r^2
The area of the outer part of the track (beyond the semi-circle) can be calculated as a rectangle with dimensions (2w + 2r) by 4.
So, the total area of the track can be calculated as:
Area of track = Area of straight sections + Area of semi-circular sections
Area of track = (4 * (2w + 2r)) + (2 * (1/2) * pi * r^2)
Simplifying further:
Area of track = 8w + 8r + pi * r^2
Therefore, the formula for the area of the track in terms of pi and r is 8w + 8r + pi * r^2.
3a) To find out how far you will run when you stay 0.5 m from the inner edge of the track, we need to calculate the perimeter of the track (the distance around the track).
The perimeter of the track consists of the straight sections along the length of the soccer field and the semi-circular sections at the ends.
Perimeter of the track = 2 * (Length of soccer field) + 2 * (Circumference of semi-circle)
The length of the soccer field is 2w as given in the problem.
The circumference of a circle is given as 2 * pi * r.
So, the perimeter of the track can be calculated as:
Perimeter of the track = 2 * (2w) + 2 * (2 * pi * r)
Simplifying further:
Perimeter of the track = 4w + 4pi * r
Therefore, when you stay 0.5 m from the inner edge of the track, you will run a distance of 4w + 4pi * r.
3b) If your friend stays 0.5 m from the outer edge of the track, we can calculate the difference in distance between you and your friend.
The difference in distance will be equal to the width of the track, which is 4 m.
Therefore, your friend will run 4 meters farther than you.