Question

Worth a 100 points!

The question is in the attachment below.

Answer 1

B. 7.5

Explanation:

• Let's solve this problem using similar triangles.

One right triangle is formed by:

• the height of the streetlight (i.e., 18 ft),

• the distance between the top of the streetlight and the top of the tree's shadow (i.e., unknown since we don't need it for the problem),
• and the distance between the base of the streetlight and the top of the tree's shadow (i.e., 15 ft between the streetlight's base and the tree's base + the unknown length of the shadow)

Another similar right triangle is formed by:

• the height of the tree (i.e., 6 ft),
• the distance between the top of the tree and the top of its shadow (i.e., also unknow since we don't need it for the problem),
• and the distance between the tree's base and the top of it's shadow (i.e., the unknown length of the shadow).

Proportionality of similar sides:

• Similar triangles have similar sides, which are proportional.
• We can use this proportionality to solve for s, the length of the tree's shadow in ft.

First set of similar sides:

• The height of the streetlight (i.e., 18 ft) is similar to the height of the tree (i.e., 6 ft).

Second set of similar sides:

• Similarly, the distance between the base of the streetlight and the top of the tree's shadow (i.e., 15 ft + unknown shadow's length) is similar to the length of the tree's shadow (i.e., an unknown length).

Now we can create proportions to solve for s, the length of the shadow:

18 / 6 = (15 + s) / s

(3 = (15 + s) / s) * s

(3s = 15 + s) - s

(2s = 15) / 2

s = 7.5

Thus, the length of the shadow is 7.5 ft.

Check the validity of the answer:

We can check our answer by substituting 7.5 for s and seeing if we get the same answer on both sides of the equation we just used to solve for s:

18 / 6 = (15 + 7.5) / 7.5

3 = 22.5 / 7.5

3 = 3

Thus, our answer is correct.

Answer 2

B. 7.5

`\hrulefill`

Explanation:

The given diagram shows two similar right triangles.

Let "x" be the base of the smaller triangle. Therefore:

• The smaller triangle has a base of x ft and a height of 6 ft.
• The larger triangle has a base of (15 + x) ft and a height of 18 ft.

In similar triangles, corresponding sides are always in the same ratio. Therefore, we can set up the following ratio of base to height:

`\begin{aligned}\sf \underline{Smaller\;triangle}\; &\;\;\;\;\;\sf \underline{Larger\;triangle}\\\\\sf base:height&=\sf base:height\\\\x:6&=(15+x):18\end{aligned}`

Express the ratios as fractions:

`(x)/(6)=((15+x))/(18)`

Cross multiply and solve for x:

`\begin{aligned}18x&=6(15+x)\\\\18x&=90+6x\\\\18x-6x&=90+6x-6x\\\\12x&=90\\\\(12x)/(12)&=(90)/(12)\\\\x&=7.5\end{aligned}`

Therefore, the shadow of the tree is 7.5 feet long.