A water storage tank has the shape of a cylinder with diameter 14 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 12 ft, what percentage of the total capacity is being used? (Round your answer to one decimal place.) %
Accepted Solution
A:
Answer:percentage of the total capacity is 75.6%Step-by-step explanation:Hello! To solve this problem we follow the following steps1. draw the complete scheme of the problem (see attached image)2. To solve this problem we must find the area of the circular sector using the following equation.(c in the second attached image)[tex]A=\frac{R^2}{2} (\alpha -sin\alpha )[/tex][tex]\alpha =2arccos(\frac{d}{R})[/tex]3. observing the attached images we replace the values in the equations and find the area of the circular sector, remember that you must transform the angle to radians[tex]\alpha =2arccos(\frac{5}{7})=88.83[/tex][tex]A=\frac{R^2}{2} (\alpha -sin\alpha )\\A=\frac{7^2}{2} (88.83-sen88.83)*\frac{\pi rad}{180} =37.57ft^2[/tex]4.we calculate the area of the total circle (At), then subtract the area of the circular sector (Ac) to find the area occupied by water (Aw)[tex]At=\frac{\pi }{4} (14ft)^2=153.93ft^2[/tex]Aw=At-Ac=153.93-37.57=116.36ft^25.Finally, we calculate the percentage that represents the water in the tank by dividing the area of the water over the total area of the tank[tex]\frac{116.36}{153.93} *100=75.6[/tex]percentage of the total capacity is 75.6%