Certainly! Let's go through each part of the question step-by-step.
### Part a: Find the inverse function
We start with the given discount formula:
[tex]\[ d = 0.15(c - 10) \][/tex]
To find the inverse function, we need to express [tex]\( c \)[/tex] (the cost) as a function of [tex]\( d \)[/tex] (the discount). We'll solve for [tex]\( c \)[/tex] in terms of [tex]\( d \)[/tex].
1. Start with the given equation:
[tex]\[ d = 0.15(c - 10) \][/tex]
2. Isolate the term involving [tex]\( c \)[/tex] by dividing both sides by 0.15:
[tex]\[ \frac{d}{0.15} = c - 10 \][/tex]
3. Simplify the division:
[tex]\[ \frac{d}{0.15} = \frac{d}{\frac{15}{100}} = \frac{100d}{15} = \frac{20d}{3} \][/tex]
So we have:
[tex]\[ \frac{20d}{3} = c - 10 \][/tex]
4. Add 10 to both sides to solve for [tex]\( c \)[/tex]:
[tex]\[ c = \frac{20d}{3} + 10 \][/tex]
Thus, the inverse function is:
[tex]\[ c(d) = \frac{20d}{3} + 10 \][/tex]
### Part b: Evaluate the inverse function for [tex]\( d = 18 \)[/tex]
We need to find the cost [tex]\( c \)[/tex] when [tex]\( d = 18 \)[/tex]:
1. Substitute [tex]\( d = 18 \)[/tex] into the inverse function:
[tex]\[ c(18) = \frac{20 \times 18}{3} + 10 \][/tex]
2. Calculate the value within the fraction first:
[tex]\[ \frac{20 \times 18}{3} = \frac{360}{3} = 120 \][/tex]
3. Add 10 to the result:
[tex]\[ c(18) = 120 + 10 = 130 \][/tex]
Thus, the cost [tex]\( c \)[/tex] for a discount of [tex]$18 is $[/tex]130.
### Part c: What was Dan's final cost for this item?
From part b, we have already calculated the cost of the item to be [tex]\( \$130 \)[/tex]. Therefore, Dan's final cost for the item was:
[tex]\[ \boxed{\$130} \][/tex]
