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A shadow 24 yards long is cast by a radio tower that is 45 yards tall. What is the height of a nearby cell phone tower that casts a shadow 32 yards long?

A shadow $24$ yards long is cast by a radio tower that is $45$ yards tall. What is the height of a nearby cell phone tower that casts a shadow $32$ yards long?

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Solve proportions: word problems

Full solution

Q. A shadow

24

yards long is cast by a radio tower that is

45

yards tall. What is the height of a nearby cell phone tower that casts a shadow

32

yards long?

Use Similar Triangles: We can use similar triangles to solve this problem. The ratio of the height of the radio tower to the length of its shadow should be the same as the ratio of the height of the cell phone tower to the length of its shadow.
Denote Heights and Ratios: Let's denote the height of the cell phone tower as $h$ . The ratio for the radio tower is $\frac{45}{24}$ , and the ratio for the cell phone tower will be $\frac{h}{32}$ .
Set Up Proportion: Now we can set up the proportion: $\frac{45}{24} = \frac{h}{32}$ .
Cross-Multiply: To find $h$ , we cross-multiply: $45 \times 32 = 24 \times h$ .
Calculate Left Side: Now we calculate the left side of the equation: $45 \times 32 = 1440$ .
Solve for h: We now have $1440 = 24 \times h$ . To solve for $h$ , we divide both sides by $24$ : $h = \frac{1440}{24}$ .
Calculate Right Side: Calculating the right side of the equation gives us $h = 60$ .

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