Question

A tree that is 3 feet tall is growing at a rate of 1 foot per year. A 5-foot tree is growing at a rate of 0.75 foot per year.

The ordered pair (t, h) represents the time in years, t, at which the trees are at height, h.

Which ordered pair represents the number of years elapsed when the trees are at the same height?



Responses

(8, 11)
begin ordered pair 8 comma 11 end ordered pair

(7, 10.25)
begin ordered pair 7 comma 10.25 end ordered pair

(6, 9)
begin ordered pair 6 comma 9 end ordered pair

(12, 14)

Answer 1

To determine the time when the two trees are at the same height, we can set up equations for the heights of the trees at time \( t \).

Let:
- The height of the 3-foot tree after \( t \) years be:
\[ h_1 = 3 + 1t = 3 + t \]

- The height of the 5-foot tree after \( t \) years be:
\[ h_2 = 5 + 0.75t \]

We want to find the time \( t \) when both trees are the same height, so we set \( h_1 \) equal to \( h_2 \):
\[ 3 + t = 5 + 0.75t \]

Now, we'll solve for \( t \):
1. Subtract \( 0.75t \) from both sides:
\[ 3 + t - 0.75t = 5 \]
\[ 3 + 0.25t = 5 \]

2. Subtract 3 from both sides:
\[ 0.25t = 2 \]

3. Divide both sides by 0.25:
\[ t = \frac{2}{0.25} = 8 \]

Now that we have \( t = 8 \), we can find the corresponding height \( h \) at that time. We can use either height equation, but let's use the first one:
\[ h = 3 + t = 3 + 8 = 11 \]

Thus, the ordered pair (t, h) representing the number of years elapsed when the trees are at the same height is:
**(8, 11)**.

So, the correct answer is:
**(8, 11)**.