A piece of rectangular sheet metal is 20 inches wide. It is to be made into a drain gutter by turning up the edges to form parallel sides. Let $x$ represent the length of each of the parallel sides.
Give the restrictions on $x$.
$0< x< 10$
(Simplify your answers. Type an integer or a decimal.)
Choose the function A that gives the area of a cross-section of a gutter.
A. $A(x)=x^{2}$
B. $A(x)=(20-2 x)^{2}$
C. $A(x)=x(20-2 x)$
D. $A(x)=x(x-20)$
The value of $x$ for which $A$ will be a maximum is $5^{\top}$.
(Simplify your answer. Type an integer or a decimal.)
The maximum area of the cross-section is 50 square inches.
(Simplify your answer. Type an integer or a decimal.)
The values of $x$ for which the area of a cross-section will be less than 30 square inches are between 0 and or between and 10. (Simplify your answers. Type an integer or a decimal. Round to the nearest hundredth if needed.)
Answer
The values of $x$ for which the area of a cross-section will be less than 30 square inches are between $0$ and $5 - \sqrt{10}$ or between $\sqrt{10} + 5$ and $10$.
Step 1 ：The restrictions on $x$ are $0 Step 2 ：The function that gives the area of a cross-section of the gutter is $A(x)=x(20-2x)$. Step 3 ：The value of $x$ that maximizes the area of the cross-section is \(\boxed{5}\). Step 4 ：The maximum area of the cross-section is \(\boxed{50}\) square inches. Step 5 ：The values of $x$ for which the area of a cross-section will be less than 30 square inches are between $0$ and $5 - \sqrt{10}$ or between $\sqrt{10} + 5$ and $10$.
