The Juice Problem. A store sells a 1-liter container of
juice for 75¢ and a 2-liter container for $1.40. Assume
that the cost varies linearly with the number of liters, and find the slope of the line that shows the relationship.
How many liters would be in a container that cost
$3.35?
First find the rule for the cost.
C(1) = 0.75
C(2) = 1.40 (difference = 0.65)
C(3) = 2.05
C(4) = ...
C(x) = mx+0.10
Find m, and then solve for
C(x) = 3.35
I am still confused, I understand finding the difference and that being 0.65. What I am not sure about is where you wrote c(x)=mx+0.10. I know that the C stands for cost, should the x be a y? I know that m stands for slope but what is the 0.10? I really do not understand linear equations.
To express the equation in a more familiar form, it could be
y=mx+b
where b is the y-intercept, meaning the value of y when x=0. In this context, it represents a fixed cost, for example, the cost of the container.
We can find b by taking the difference of C(1)-m, or 0.75-0.65 = 0.10.
We therefore end up with
y=0.65x + 0.10
So the question reduces to when y=3.35, solve for x in
3.35=0.65x + 0.10
Okay that clears it up. Thank you very much for your help.
You're welcome!
To find the slope of the line that shows the relationship between the cost of juice and the number of liters, we need to determine the rate of change in cost per liter.
First, we can calculate the cost per liter for the 1-liter container:
Cost per liter = 75¢ / 1 liter = 75¢.
Next, we can calculate the cost per liter for the 2-liter container:
Cost per liter = $1.40 / 2 liters = 70¢.
Now we have two data points: (1, 75¢) and (2, 70¢), where the x-values represent the number of liters and the y-values represent the cost per liter.
The slope of the line can be found using the formula:
slope = (change in y) / (change in x)
Let's calculate the change in y:
change in y = 70¢ - 75¢ = -5¢.
Now, let's calculate the change in x:
change in x = 2 liters - 1 liter = 1 liter.
Plugging these values into the slope formula:
slope = (-5¢) / (1 liter) = -5¢/liter.
Therefore, the slope of the line that shows the relationship between the cost of juice and the number of liters is -5¢/liter.
To determine how many liters would be in a container that costs $3.35, we can use the slope-intercept form of a linear equation (y = mx + b), where y represents the cost, x represents the number of liters, m represents the slope, and b represents the y-intercept.
Substituting the known information into the equation:
$3.35 = (-5¢/liter) * x + b.
Now, we need to find the value of b, which is the y-intercept. We can do this by plugging in one of the known data points, such as (1, 75¢):
75¢ = (-5¢/liter) * 1 liter + b.
Simplifying this equation, we find:
75¢ = -5¢ + b,
b = 80¢.
Now we have the equation:
$3.35 = (-5¢/liter) * x + 80¢.
To find the number of liters (x) that costs $3.35, we can rearrange and solve for x:
$3.35 - 80¢ = (-5¢/liter) * x,
$2.55 = (-5¢/liter) * x.
Dividing both sides by -5¢:
($2.55 / -5¢) = x,
-51 liters = x.
Therefore, a container that costs $3.35 would have a volume of -51 liters. However, since it doesn't make sense to have a negative volume, we can conclude that there is an error in the calculation. Please double-check the given information or calculations.