Question

A 14 ft tall tent standing next to a cardboard box castsa 40 ft shadow. If the cardboard box casts a shadow that is 70 ft long then how tall is it?. Show all your work to earn full credit

Answer 1

To find the height of the cardboard box, the concept of similar triangles is used. The proportion of the tent's height to its shadow is equal to the proportion of the box's height to its shadow, which calculates to a height of 24.5 feet for the box.

Step-by-step explanation:

To determine the height of the cardboard box, we can use the properties of similar triangles. The tent and its shadow form one triangle, and the cardboard box and its shadow form another. Since the angles of these triangles are the same (assuming the ground is level and the sun's rays are coming in at the same angle for both the tent and the box), the triangles are similar, and the ratios of their corresponding sides are equal.

Let x be the height of the cardboard box. We can set up the following proportion using the tent and its shadow as the first ratio and the cardboard box and its shadow as the second ratio:

\( \frac{14\text{ ft}}{40\text{ ft}} = \frac{x}{70\text{ ft}} \)

Now we solve for x:

1. Multiply both sides by 70 ft to isolate x on one side:
   \( 70\text{ ft} \times \frac{14\text{ ft}}{40\text{ ft}} = x \)
2. Calculate the right side:
   \( x = 70\text{ ft} \times \frac{14}{40} = 24.5\text{ ft} \)

Therefore, the height of the cardboard box is 24.5 feet.