Question

Caroline wants to create a fun adventure called "Zoo Explorers." This adventure will have riddles and stops that guide visitors around the zoo. She wants to strike a balance: enough stops for fun but not so many that visitors can't finish in one day.

Without any adventure stops in the zoo, visitors spend an average of 5 hours at the zoo. In a trial run of "Zoo Explorers" with 4 adventure stops, participants stayed for an average of 6 total hours.
Which linear equation represents the scenario in #2?

y=4x+6

y=
4
1
​
x+5

y=
2
1
​
x+3

y=5x

Answer 1

Let's define the variables first:

- Let \( x \) be the number of adventure stops.
- Let \( y \) be the total time spent at the zoo in hours.

From the information given:

- Without any stops (when \( x = 0 \)), visitors spend an average of 5 hours, so when \( x = 0 \), \( y = 5 \).
- With 4 stops (when \( x = 4 \)), visitors spend an average of 6 hours, so when \( x = 4 \), \( y = 6 \).

We can use these two points to find the equation of the line:
1. The point (0, 5) indicates that the y-intercept is 5.
2. The point (4, 6) can help determine the slope.

Calculating the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 5}{4 - 0} = \frac{1}{4}
\]

Using the point-slope formula \( y - y_1 = m(x - x_1) \), applying the y-intercept (0, 5) gives us:
\[
y - 5 = \frac{1}{4}(x - 0)
\]
\[
y = \frac{1}{4}x + 5
\]

This matches the equation \( y = \frac{1}{4}x + 5 \). So the correct answer is:

**\( y = \frac{1}{4}x + 5 \)**.