Question

A sealed bottle containing some water is made up of two cylinders \[{\rm A}\] and \[{\rm B}\] of radius \[1.5cm\] and \[3cm\] respectively, as shown in the figure. When the bottle is placed right up on a table, the height of water in it is \[15cm\] , but when placed upside down, the height of water is \[24cm\] . The height of the bottle is

A. \[25cm\]
B. \[26cm\]
C. \[27cm\]
D. \[28cm\]

Answer 1

Hint: In this problem, we need to convert this word problem into mathematical expression. Here, the sealed bottle containing some water made up of two cylinders of radius is given in the problem. When the bottle is placed upside down on the table the notice the height of the water is given then finally find the height of the bottle by using the volume of cylinder formula is \[V = \pi {r^2}h\] .
Where, \[r - \] radius of the cylinder and height of the cylinder, \[H + h\] .

Complete step by step solution:
In the given problem,
Let the two cylinder be \[{\rm A}\] and \[{\rm B}\]
Radius of \[{\rm A}\] is \[r = 1.5cm\] and Radius of \[{\rm B}\] is \[r = 3cm\]

To find the height of the cylinder, we get
Total height \[ = H + h\]
The formula for finding the volume of cylinder, \[V = \pi {r^2}h\]
The first sealed bottle has two cylinder shape is equal to the second inverted cylinder, we get
 \[\pi {(2r)^2}H + \pi {r^2}(15 - H) = \pi {r^2}h + \pi {(2r)^2}(24 - h)\]
Take out \[\pi {r^2}\] on both sides, we get
 \[\pi {r^2}(4H + 15 - H) = \pi {r^2}(h + 4(24 - h))\]
By dividing on both sides by \[\pi {r^2}\] , we can get
 \[3H + 15 = h + 96 - 4h\]
By simplify the arithmetic operation, we get
 \[3H + 15 = 96 - 3h\]
To simplify in further step, we have
 \[
  3(H + 5) = 3(32 - h) \\
  H + h = 32 - 5 = 27 \;
 \]
So, now we get
Total height, \[H + h = 27cm\]
Therefore, the height of the cylinder is \[27cm\]
So, the correct answer is “ \[27cm\] ”.

Note: We note that the cylinder shaped sealed bottle has filled with water.so, the height of water is mention and we have to find the total height of the cylinder by using the volume of the cylinder formula is \[V = \pi {r^2}h\] .we need to remind the formula to calculate this sort of problem. We have to read the question carefully and then convert it into a mathematical expression then it will be solved by using a related formula.