Ethan bought 3 pints of raspberries for $[tex]$5$[/tex] per pint and [tex]$x$[/tex] pints of blueberries for $[tex]$3$[/tex] per pint. The average cost of all the berries he bought was [tex]$\frac{15+3 x}{3+x}$[/tex] dollars per pint. If the equation [tex]$y=\frac{15+3 x}{3+x}$[/tex] is graphed in the [tex]$xy$[/tex]-plane, what quantity will be represented by the [tex]$y$[/tex]-intercept of the graph?
A. The total cost, in dollars, of all the raspberries Ethan bought
B. The average cost, in dollars per pint, of all the raspberries Ethan bought
C. The total cost, in dollars, of all the blueberries Ethan bought
D. The average cost, in dollars per pint, of all the blueberries Ethan bought
Answer (1)
Find the y-intercept by setting x = 0 in the equation y = 3 + x 15 + 3 x .
Calculate the value of y when x = 0 : y = 3 + 0 15 + 3 ( 0 ) = 3 15 = 5 .
Interpret the y-intercept in the context of the problem: it represents the average cost per pint when only raspberries are bought.
The y-intercept represents the average cost, in dollars per pint, of all the raspberries Ethan bought: T h e a v er a g ecos t , in d o ll a rs p er p in t , o f a llt h er a s p b err i es Et hanb o ug h t
Explanation
Understanding the Problem We are given the equation y = 3 + x 15 + 3 x , which represents the average cost of berries in dollars per pint. We need to find the y-intercept of this graph and interpret its meaning in the context of the problem.
Finding the y-intercept To find the y-intercept, we set x = 0 in the equation y = 3 + x 15 + 3 x . This is because the y-intercept is the point where the graph intersects the y-axis, which occurs when x = 0 .
Calculating the y-intercept Substituting x = 0 into the equation, we get: y = 3 + 0 15 + 3 ( 0 ) = 3 + 0 15 + 0 = 3 15 = 5
Interpreting the y-intercept The y-intercept is 5. Now we need to interpret what this value represents. Since x represents the number of pints of blueberries, setting x = 0 means Ethan only bought raspberries. The y-value represents the average cost per pint when only raspberries are bought.
Verifying the Interpretation Ethan bought 3 pints of raspberries for $5 per pint. The total cost of the raspberries is 3 × 5 = $15 . The average cost per pint of raspberries is 3 15 = $5 per pint. This matches the y-intercept value we calculated.
Final Answer Therefore, the y-intercept of the graph represents the average cost, in dollars per pint, of all the raspberries Ethan bought. The final answer is T h e a v er a g ecos t , in d o ll a rs p er p in t , o f a llt h er a s p b err i es Et hanb o ug h t
Examples
Imagine you're running a fruit stand. You sell apples at $2 each and oranges at $1 each. If you want to know the average price per fruit when you only sell apples (no oranges), finding the y-intercept is like figuring out that average price. In this case, it's simply the price of an apple, which is $2 . This concept helps you understand your pricing strategy and how it changes as you introduce different products.
