Question

lucia is renting a stand - up paddleboard from sunrise lake rentals. she is charged an initial rental fee and an additional cost per hour she rents the paddleboard. this situation can be modeled as a linear relationship.

Explanation:

To solve this problem, we can determine the linear equation representing the total cost of renting the paddleboard. Let's assume we want to find the cost per hour (slope) and the initial fee (y - intercept).

Step 1: Identify two points on the line

From the graph, we can see that when \( x = 0 \) (0 hours), the total cost \( y=\$10 \) (initial fee). When \( x = 5 \) hours, the total cost \( y = \$70 \). So the two points are \( (0, 10) \) and \( (5, 70) \).

Step 2: Calculate the slope (cost per hour)

The slope \( m \) of a line passing through two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is given by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \)

Substituting \( (x_1,y_1)=(0, 10) \) and \( (x_2,y_2)=(5, 70) \) into the formula:
\( m=\frac{70 - 10}{5-0}=\frac{60}{5} = 12 \)

So the cost per hour is \( \$12 \) per hour.

Step 3: Determine the equation of the line

The equation of a line in slope - intercept form is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

We know that \( m = 12 \) and from the point \( (0,10) \), when \( x = 0 \), \( y=b \), so \( b = 10 \)

The equation of the line is \( y=12x + 10 \)

If we want to find the total cost for a certain number of hours, we can substitute the value of \( x \) (number of hours) into this equation. For example, if we want to find the cost for 3 hours:

Step 4: Substitute \( x = 3 \) into the equation

\( y=12\times3+ 10=36 + 10=\$46 \)

If the question was to find the cost per hour, the answer is \(\$12\) per hour. If the question was to find the initial fee, the answer is \(\$10\). If the question was to find the total cost for a certain number of hours, we can use the equation \( y = 12x+10 \) to calculate it.

For example, if we assume the question is "What is the cost per hour of renting the paddleboard?"

Answer:

The cost per hour of renting the paddleboard is \(\$12\)