Optimisation problems Quiz

1. A rectangle has perimeter 20 m and sides $x$ and $10-x$. What is the maximum area?

• A. $20$ m$^2$
• B. $25$ m$^2$
• C. $50$ m$^2$
• D. $100$ m$^2$

2. A closed cylinder has fixed volume $V$. To minimise its surface area $S=2\pi r^2+2\pi rh$ you should first

• A. differentiate with respect to both $r$ and $h$
• B. set $S$ equal to $V$
• C. use the volume to eliminate $h$, so $S$ is a function of $r$ only
• D. ignore the volume and differentiate $S$ as given

3. The profit is $P=-2x^2+40x-50$. What is the maximum profit?

• A. $150$
• B. $200$
• C. $100$
• D. $-50$

4. Squares of side $x$ cm are cut from the corners of a $12$ cm square card to make an open box, with $V=x(12-2x)^2$. The sensible domain is

• A. $0<x<12$
• B. $0<x<4$
• C. $0<x<3$
• D. $0<x<6$

5. The surface area of a solid is $S=x^2+\frac{16}{x}$ for $x>0$. What is its minimum value?

• A. $8$
• B. $20$
• C. $12$
• D. $16$