Question

floodlights
eliot is playing football. he is 6 feet tall. he stands exactly half way between two floodlights. the floodlights are 12 yards high and 50 yards apart. the floodlights give eliot two shadows, falling in opposite directions. 1 yard is 3 feet.

1. draw a diagram to represent this situation. label your diagram with the measures.
2. find the total length of eliots shadows. explain your reasoning in detail.
3. suppose eliot walks in a straight line towards one of the floodlights.

figure out what happens to the total length of eliots shadows. explain your reasoning in detail.

Explanation:

Step1: Convert units

Since \(1\) yard \( = 3\) feet, the height of the flood - light \(h_f=12\times3 = 36\) feet, and the distance between the two flood - lights \(d = 50\times3=150\) feet. Eliot's height \(h_e = 6\) feet, and the distance from Eliot to each flood - light \(x=\frac{150}{2}=75\) feet.

Step2: Use similar triangles

Let the length of one shadow be \(s_1\) and the other be \(s_2\).
For the similar triangles formed by Eliot and the flood - light, we have the proportion \(\frac{h_e}{h_f}=\frac{s_1}{s_1 + x}\) (using the left - hand side similar triangles, and the same proportion holds for the right - hand side).
Substitute \(h_e = 6\), \(h_f = 36\), and \(x = 75\) into \(\frac{6}{36}=\frac{s}{s + 75}\).
Cross - multiply: \(6(s + 75)=36s\).
Expand: \(6s+450 = 36s\).
Subtract \(6s\) from both sides: \(450=36s - 6s\).
Simplify: \(30s = 450\), so \(s = 15\) feet.

Step3: Calculate the total length of the shadows

The total length of the two shadows \(L=s_1 + s_2\). Since \(s_1=s_2 = 15\) feet (by symmetry), \(L=15 + 15=30\) feet.

Answer:

The total length of Eliot's shadows is \(30\) feet.