If i want to compare the length of a shadow cast by a basketball goalpost 10ft with 2ft backboard shadow 69 inches to the shadow of the goalpost measured with a yardstick the shadow is 18inches and i get 3ft 6in for the goalpost & backboard can I say that the height of the basketball goals are similar or are they the same based on equal ratios?

Question

If i want to compare the length of a shadow cast by a basketball goalpost 10ft with 2ft backboard shadow 69 inches to the shadow of the goalpost measured with a yardstick the shadow is 18inches and i get 3ft 6in for the goalpost & backboard can I say that the height of the basketball goals are similar or are they the same based on equal ratios?

Answer 1

To determine if the height of the basketball goals are similar or the same based on equal ratios, we need to compare the ratios of the shadows to the actual heights of the goalposts.

First, let's calculate the ratio of the shadow of the goalpost to its actual height using the first measurement:

Shadow length: 18 inches
Actual height: 3 feet (36 inches)

Ratio = Shadow length / Actual height = 18 / 36 = 1/2

Now, let's calculate the ratio of the shadow of the goalpost with the backboard to its actual height using the second measurement:

Shadow length: 69 inches
Actual height: 3 feet 6 inches (42 inches)

Ratio = Shadow length / Actual height = 69 / 42

Simplifying this ratio is important to compare it with the previous ratio. The simplified ratio will be:

Ratio = 23 / 14

Now, let's compare the two ratios:
1/2 and 23/14

We can say that if the two ratios are equal, then the height of the basketball goals are the same. If the ratios are different, the height of the goals are simply similar, but not equal.

To determine if the ratios are equal, we can cross-multiply:

Cross-multiplying the ratios:

1/2 = 23/14
14 * (1/2) = 2 * 23
7 = 46

Since 7 is not equal to 46, we can conclude that the ratios are not equal. Therefore, the height of the basketball goals is similar, but not the same.