Mensuration - Volume of Cube, Cuboid, and Cylinder

A cuboidal water tank is $6 m$ long, $5 m$ wide, and $4.5 m$ deep. How many liters of water can it hold?

Step 1: Identify the given dimensions: length $l = 6 m$, breadth $b = 5 m$, and height $h = 4.5 m$. Step 2: Use the formula for the volume of a cuboid: $V = l \times b \times h$. Step 3: Substitute the values: $V = 6 \times 5 \times 4.5 = 135 m^3$. Step 4: Convert the volume from cubic meters to liters. We know that $1 m^3 = 1000 L$. Step 5: Total capacity in liters = $135 \times 1000 = 135,000 L$.

To find the capacity, we first calculate the volume in cubic meters by multiplying the length, width, and depth, then convert the result to liters using the standard conversion factor.

Find the height of a cylinder whose volume is $1.54 m^3$ and the diameter of the base is $140 cm$.

Step 1: Convert all units to meters. Volume $V = 1.54 m^3$. Diameter $d = 140 cm = 1.4 m$. Radius $r = \frac{d}{2} = \frac{1.4}{2} = 0.7 m$. Step 2: Use the formula for the volume of a cylinder: $V = \pi r^2 h$. Step 3: Substitute the known values ($\pi = \frac{22}{7}$): $1.54 = \frac{22}{7} \times (0.7)^2 \times h$. Step 4: Simplify the expression: $1.54 = \frac{22}{7} \times 0.49 \times h$. Step 5: Further simplify: $1.54 = 22 \times 0.07 \times h \Rightarrow 1.54 = 1.54 \times h$. Step 6: Solve for $h$: $h = \frac{1.54}{1.54} = 1 m$.

The problem requires finding the height given volume and diameter. Consistency in units is key, so we converted centimeters to meters before applying the volume formula for a cylinder.

Volume of first cube $V_1 = 3^3 = 27 cm^3$ Volume of second cube $V_2 = 4^3 = 64 cm^3$ Volume of third cube $V_3 = 5^3 = 125 cm^3$ Total volume of new cube $V = V_1 + V_2 + V_3$ $V = 27 + 64 + 125 = 216 cm^3$ Let the edge of the new cube be $A$. $A^3 = 216$ $A = \sqrt[3]{216} = 6 cm$

Since the cubes are melted and recast, the total volume remains conserved. We calculate the volumes of the three smaller cubes, sum them up to find the volume of the large cube, and then find its cube root to determine the edge length.

$\text{Radius (r)} = 14\text{ cm}$ $\text{Height (h)} = 3\text{ m} = 300\text{ cm}$ $\text{Volume of one cylinder} = \pi r^2 h$ $\text{Volume} = \frac{22}{7} \times 14 \times 14 \times 300$ $\text{Volume} = 22 \times 2 \times 14 \times 300$ $\text{Volume} = 44 \times 4200$ $\text{Volume} = 184800\text{ cm}^3$

To find the volume of a cylinder, we use the formula $V = \pi r^2 h$. First, ensure all units are consistent by converting the height from meters to centimeters ($1\text{ m} = 100\text{ cm}$). Substituting the values $r = 14$ and $h = 300$ into the formula allows us to calculate the space occupied by the pillar.