A billboard is 2.5 times as long as it is wide. The area of the actual billboard is 2,250 square feet. A scale drawing is made of the billboard and the area of the scale is 160 square inches. What is the scale used in the scale drawing?
the area grows as the square of the linear scale. So the scale is
√(160in^2/2250ft^2)
= √(160/32400 in^2)
= √(1/2025)
= 1:45
This is a completely incorrect answer to this problem.
To find the scale used in the scale drawing, we need to compare the dimensions of the actual billboard to the dimensions of the scale drawing.
Let's define:
- Length of the actual billboard as L (in feet)
- Width of the actual billboard as W (in feet)
- Length of the scale drawing as l (in inches)
- Width of the scale drawing as w (in inches)
Given:
- The length of the billboard is 2.5 times its width.
So, L = 2.5W
- The area of the actual billboard is 2,250 square feet.
So, L * W = 2,250
- The area of the scale drawing is 160 square inches.
So, l * w = 160
We can solve these equations simultaneously to find the values of L, W, l, and w.
1. Using the first equation (L = 2.5W), we can express L in terms of W:
L = 2.5W
2. Substituting L from the first equation into the second equation (L * W = 2,250):
(2.5W) * W = 2,250
2.5W^2 = 2,250
3. Dividing both sides of the equation by 2.5:
W^2 = 900
4. Taking the square root of both sides:
W = √900
W = 30
Now we know that the width of the actual billboard (W) is 30 feet.
5. Substituting the value of W (30) into the first equation (L = 2.5W):
L = 2.5 * 30
L = 75
So, the length of the actual billboard (L) is 75 feet.
Now let's find the values of l and w.
6. Using the third equation, we can express w in terms of l:
l * w = 160
w = 160 / l
7. Substituting L and W into the fourth equation (l * w = 160):
75 * 30 = 160 * w
2,250 = 160w
8. Dividing both sides of the equation by 160:
w = 2,250 / 160
w ≈ 14.06
Now we know that the width of the scale drawing (w) is approximately 14.06 inches.
9. Substituting the value of w (14.06) into the sixth equation (w = 160 / l):
14.06 = 160 / l
10. Solving for l:
l = 160 / 14.06
l ≈ 11.37
So, the length of the scale drawing (l) is approximately 11.37 inches.
Finally, to find the scale used in the scale drawing, we compare the dimensions of the actual billboard to the dimensions of the scale drawing:
Scale = (length of the scale drawing / length of the actual billboard) = l / L = 11.37 / 75.
Therefore, the scale used in the scale drawing is approximately 0.1516, or you can say the scale is approximately 1:6.6.