Question

At a bowling alley, shoe rentals cost $3 and each game costs $4. The average cost c (in dollars) of bowling n games is given by c=4n+3n. Find how many games you must bowl for the average cost to fall to $4.75 by (a) solving an equation, and (b) using the inverse of a function.

Answer 1

To find the number of games you must bowl for the average cost to fall to $4.75, you can either solve the equation or use the inverse of a function. The answer is approximately 0.6786 or 1 game, which would be rounded up.

Step-by-step explanation:

To find the number of games you must bowl for the average cost to fall to $4.75, let's set up the equation:

c = 4n + 3n

Given that the average cost is $4.75, we can substitute this value into the equation:

4.75 = 4n + 3n

Combine like terms:

4.75 = 7n

Divide both sides by 7 to solve for n:

n = 4.75/7 = 0.6786

So, you must bowl approximately 0.6786 or 1 game for the average cost to fall to $4.75.

Keep in mind that you cannot bowl a fraction of a game, so the answer would be rounded up to 1 game.

To use the inverse of a function, we need to solve 4n + 3n = 4.75 for n:

7n = 4.75

n = 4.75/7

= 0.6786

Rounding up, you would need to bowl 1 game for the average cost to fall to $4.75.

Answer 2

Answer: 0.67857

Explanation:

a ) Given the equation we simply group terms and solve n for c =4.75:

`c = 4n+3n\\c = n(4+3)\\c/7 =n=>4.75/7 = 0.678\\\\`

b) We first check if the function is invertible (let n1 and n2 be any values for the function c = 4n+3n) So we have to prove (
`f(n_(1)) = f(n_(2))` =>
`n_(1) = n_(2)`)

so we have
`f(n_(1))= 7n_(1)\\f(n_(2)) = 7n_(2)\\  ( f(n_(1)) =f(n_(2)) ) =(7n_(1)=7n_(2)) = (n_(1)=n_(2))`

finally
`n_(1) = n_(2)`. This is a check you always do to find if a function is invertible (it has to be injective).

Now solving the problem we just replace c with n and n with c:

`n = 4c + 3c = c(4+3) = 7c => n/7 = c`

where n = 4.75

finally:

`c = 4.75/7 = 0.67857`

that is the number of games you need.