Math

QuestionRideEm Bicycles can make 170 bikes for \$10,300 and 190 bikes for \$10,900.
(a) Find the cost function C(x)=C(x)=. (b) What are the fixed costs in dollars? (c) What are the variable costs in dollars?

Studdy Solution

STEP 1

Assumptions1. The cost function is linear. . The cost function is in the form of C(x)=mx+bC(x) = mx + b, where mm is the variable cost per bicycle (slope of the line), bb is the fixed cost (y-intercept), and xx is the number of bicycles.
3. The factory can produce170 bicycles at a total cost of 10,300.<br/>4.Thefactorycanproduce190bicyclesatatotalcostof10,300.<br />4. The factory can produce190 bicycles at a total cost of 10,900.

STEP 2

We can find the variable cost per bicycle (slope of the line) by using the formula for the slope of a line, which is given by the change in y divided by the change in x. In this case, the change in y is the change in total cost and the change in x is the change in the number of bicycles.
m=ΔyΔxm = \frac{\Delta y}{\Delta x}

STEP 3

Plug in the given values for the change in total cost and the change in the number of bicycles to calculate the slope.
m=($10,900$10,300)(190170)m = \frac{(\$10,900 - \$10,300)}{(190 -170)}

STEP 4

Calculate the slope.
m=$60020=$30m = \frac{\$600}{20} = \$30

STEP 5

Now that we have the variable cost per bicycle, we can find the fixed cost by rearranging the equation for the cost function to solve for bb.
b=ymxb = y - mx

STEP 6

We can use either of the given points to find the fixed cost. Let's use the point (170, 10,300).Pluginthegivenvaluesfor10,300). Plug in the given values for y,, m,and, and xtocalculate to calculate b$.
b=$10,300$30×170b = \$10,300 - \$30 \times170

STEP 7

Calculate the fixed cost.
b=$10,300$5,100=$5,200b = \$10,300 - \$5,100 = \$5,200

STEP 8

Now that we have the variable cost per bicycle and the fixed cost, we can write the cost function.
C(x)=$30x+$5,200C(x) = \$30x + \$5,200(a) The linear function that models the total cost for RideEm to produce xx bicycles is C(x)=$30x+$5,200C(x) = \$30x + \$5,200.
(b) The company's daily fixed costs are $5,200 dollars.
(c) RideEm's variable costs are $30 dollars per bicycle.

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