To solve the problem of finding the equation that represents the distance [tex]\( y \)[/tex] from the lighthouse based on the number of hours [tex]\( x \)[/tex], we approach as follows:
First, we recognize that these data points should fit the equation of a straight line [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
The slope [tex]\( m \)[/tex] can be calculated using the following steps:
1. Choose two points from the table. Let's pick the points (2, 53) and (4, 95.5).
2. The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Substituting the given points:
[tex]\[
m = \frac{95.5 - 53}{4 - 2} = \frac{42.5}{2} = 21.25
\][/tex]
Next, we determine the y-intercept [tex]\( c \)[/tex]:
1. Using the slope [tex]\( m = 21.25 \)[/tex] and one of the points, say (2, 53), we apply the formula for [tex]\( c \)[/tex]:
[tex]\[
y = mx + c
\][/tex]
substituting [tex]\( x = 2 \)[/tex] and [tex]\( y = 53 \)[/tex]:
[tex]\[
53 = 21.25(2) + c
\][/tex]
2. Solving for [tex]\( c \)[/tex]:
[tex]\[
53 = 42.5 + c
\][/tex]
[tex]\[
c = 53 - 42.5
\][/tex]
[tex]\[
c = 10.5
\][/tex]
Thus, the equation that represents the distance [tex]\( y \)[/tex] from the lighthouse based on the number of hours [tex]\( x \)[/tex] is:
[tex]\[
y = 21.25 x + 10.5
\][/tex]
Comparing this with the given options, we find:
- Option A: [tex]\( y = 42.5 x + 10.5 \)[/tex]
- Option B: [tex]\( y = 10.5 x + 32 \)[/tex]
- Option C: [tex]\( y = 21.25 x + 10.5 \)[/tex]
- Option D: [tex]\( y = 12.25 x + 28.5 \)[/tex]
- Option E: [tex]\( y = 12.5 x + 10.5 \)[/tex]
Therefore, the correct answer is:
C. [tex]\( y = 21.25 x + 10.5 \)[/tex]
