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10 problems match “surface area”. Clear

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1. Total surface area of a pentagonal prism ●○○○○ 10 ptsA regular pentagonal prism has a base side length of 6 centimeters, a base apothem of 4 centimeters, and a height of 5 centimeters. Find its total surface area in square centimeters.
2. Surface area from edge sum and diagonal ●●○○○ 20 ptsThe sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. What is the total surface area of the box?
3. Minimum surface area of a square prism ●●○○○ 20 ptsA closed rectangular prism has a square base with side length $L$ and height $H$. Its volume is $10000$. Find the minimum possible surface area. Round to three decimal places.
4. Open-top box with minimum surface area ●●○○○ 20 ptsA container company makes an open-top rectangular box with a square base. The box must have a volume of $108\ \text{cm}^3$. Let the side length of the square base be $x$ cm and the height be $h$ cm. (1) Find the value of $x$ that gives the minimum outer surface area. (2) Find the minimum surface area, in $\text{cm}^2$.
5. Gallons of paint for cylindrical columns ●●○○○ 20 ptsTony will paint 16 right cylindrical columns. The top and bottom faces of each column will be covered, so those faces will not be painted. Each column is 18 feet tall and has a diameter of 10 feet. One gallon of paint covers 350 square feet. Paint is sold only in full gallons. How many gallons of paint must Tony buy to paint all 16 columns?
6. Bricks painted by a container ●●○○○ 20 ptsA container has enough paint to cover an area of $9.375$ m$^2$. Each brick measures $22.5$ cm by $10$ cm by $7.5$ cm, and its whole surface is painted. How many such bricks can be painted with the paint in the container?
7. Bricks that a container of paint covers ●●○○○ 20 ptsA container of paint is enough to paint an area of $9.375\ \text{m}^2$. Each brick measures $22.5\ \text{cm} \times 10\ \text{cm} \times 7.5\ \text{cm}$. Assuming the whole surface of each brick is painted, how many bricks can be painted with this paint?
8. Cuboid of minimal surface area ●●●○○ 50 ptsA juice box is a rectangular box (cuboid) with volume $200\ \text{cm}^3$ ($200$ ml). Find the length $l$, width $w$ and height $h$ (in cm) of the box with the least possible surface area. Round each answer to five decimal places.
9. Minimal surface area for fixed volume ●●●○○ 50 ptsA regular right pentagonal prism has a volume of $278.3105$ cubic cm. Let $x$ be the side length of the regular pentagonal bases and $y$ the length (height) of the prism, in cm. Find the values of $x$ and $y$ that minimize the total surface area. Use the area of a regular pentagon $\approx 1.720477\,x^2$. Round each value to four decimal places.
10. Longest edge of a rectangular block ●●●○○ 50 ptsThe lengths of three unequal edges of a rectangular solid block are in geometric progression. The volume of the block is $216\ \text{cm}^3$ and its total surface area is $252\ \text{cm}^2$. Find the length of the longest edge, in cm.