Suppose the demand for a certain brand of a product is given by \( D(p)=\frac{-p^{2}}{352}+50 \), where \( p \) is the price in dollars. If the price, in terms of the cost \( c \), is expressed as \( p(c)=4 c-16 \), find the demand function in terms of the cost.

To find the demand function in terms of cost, we start with the given demand function: \[ D(p) = \frac{-p^2}{352} + 50 \] Next, we substitute the expression for price \( p(c) = 4c - 16 \) into the demand function. First, we calculate \( p^2 \): \[ p^2 = (4c - 16)^2 = 16c^2 - 128c + 256 \] Now we can substitute \( p^2 \) back into the demand function: \[ D(c) = \frac{-(16c^2 - 128c + 256)}{352} + 50 \] Now, we simplify it: \[ D(c) = \frac{-16c^2 + 128c - 256}{352} + 50 \] To combine the terms, we can convert \( 50 \) to have a common denominator with \( 352 \): \[ 50 = \frac{50 \times 352}{352} = \frac{17600}{352} \] Thus, we rewrite the demand function: \[ D(c) = \frac{-16c^2 + 128c - 256 + 17600}{352} \] \[ D(c) = \frac{-16c^2 + 128c + 17344}{352} \] Finally, we can simplify this expression. Factoring it out for clarity, we write: \[ D(c) = \frac{-16(c^2 - 8c + 1084)}{352} \] And simplifying further gives us: \[ D(c) = \frac{-c^2 + 8c + 1084}{22} \] Putting it all together: \[ D(c) = \frac{-c^2 + 8c + 1084}{22} \] And there we have the demand function in terms of cost \( c \).