Directions (58-60): Read the information carefully and answer the following question given below. A right circular cylindrical vessel was filled with milk and water in the ratio 4: 5. If x ml of mixture was sold and replaced with x ml of pure milk , then the ratio of milk to water in the final mixture becomes 57:60.The quantity of mixture sold is 108 ml less than the initial quantity of mixture in the vessel.

A right circular cylindrical vessel was filled with milk and water in the ratio 4: 5. If x ml of mixture was sold and replaced with x ml of pure milk , then the ratio of milk to water in the final mixture becomes 57:

If the radius of vessel is 0.7 cm and 37 cm³ of the vessel is empty, then find the height of the vessel.

Step-by-step Explanation:

Step 1: Understand the initial ratio and volumes
Let the initial volume of the mixture in the vessel be $V$ ml.
The initial ratio of milk to water in the vessel is $4:5$.
When $x$ ml of this mixture is sold, the ratio of milk to water in the remaining mixture $(V-x)$ ml remains $4:5$.

Therefore, in the remaining mixture:
Volume of milk =
$\frac{4}{9}(V-x)$
Volume of water =
$\frac{5}{9}(V-x)$

Step 2: Add pure milk and set up the ratio equation
When $x$ ml of pure milk is added to the remaining mixture, the new quantities of milk and water become:
New volume of milk =
$\frac{4}{9}(V-x)+x=\frac{4V-4x+9x}{9}=\frac{4V+5x}{9}$
New volume of water =
$\frac{5}{9}(V-x)$

The final ratio of milk to water is given as $57:60$ (which simplifies to $19:20$). We can write:
$\frac{\frac{4V+5x}{9}}{\frac{5(V-x)}{9}}=\frac{19}{20}$

Simplifying the fraction:
$\frac{4V+5x}{5(V-x)}=\frac{19}{20}$
Cross-multiplying by 5 on both sides gives:
$\frac{4V+5x}{V-x}=\frac{19}{4}$

Solving for $V$ and $x$:
$4(4V+5x)=19(V-x)$
$16V+20x=19V-19x$
$39x=3V$
$V=13x$

Step 3: Solve for the initial volume
We are given that the quantity sold ($x$) is 108 ml less than the initial quantity ($V$):
$x=V-108$

Substitute $V=13x$ into the equation:
$x=13x-108$
$12x=108$
$x=9\text{ml}$

So, the initial quantity of mixture in the vessel is:
$V=13\times 9=117{\text{cm}}^3$ (since 1 ml = 1 cm³).

Step 4: Find the total volume and height of the vessel
The vessel is filled with 117 cm³ of mixture, and 37 cm³ of the vessel remains empty.
Total volume capacity of the vessel =
$117+37=154{\text{cm}}^3$

The volume of a right circular cylinder is given by the formula:
$\text{Volume}=\pi r^{2}h$

Given the radius $r=0.7\text{cm}$ and using $\pi =\frac{22}{7}$:
$154=\frac{22}{7}\times {\left(0.7\right)}^2\times h$
$154=\frac{22}{7}\times 0.49\times h$
$154=22\times 0.07\times h$
$154=1.54\times h$
$h=\frac{154}{1.54}=100\text{cm}$

Therefore, the height of the vessel is 100 cm.