We are given two cases of conical vessels, and we need to find their capacities in liters.
The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the base and \( h \) is the height of the cone. Since the volume is typically given in cubic centimeters (cm³) for smaller measurements, and we need the capacity in liters, we know that: \[ 1 \, \text{liter} = 1000 \, \text{cm}^3 \] So, after calculating the volume in cm³, we will divide by 1000 to convert to liters.
We are given the slant height \( l = 25 \, \text{cm} \) and the radius \( r = 7 \, \text{cm} \). To find the height \( h \), we use the Pythagorean theorem: \[ l^2 = r^2 + h^2 \] Substituting the known values: \[ 25^2 = 7^2 + h^2 \] \[ 625 = 49 + h^2 \] \[ h^2 = 625 - 49 = 576 \] \[ h = \sqrt{576} = 24 \, \text{cm} \] Now, we can calculate the volume: \[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (7^2) (24) \] \[ V = \frac{1}{3} \times 3.14 \times 49 \times 24 = 1231.36 \, \text{cm}^3 \] Converting to liters: \[ V = \frac{1231.36}{1000} = 1.231 \, \text{liters} \]
In this case, we are given the height \( h = 12 \, \text{cm} \) and the slant height \( l = 13 \, \text{cm} \). To find the radius \( r \), we use the Pythagorean theorem: \[ l^2 = r^2 + h^2 \] Substituting the known values: \[ 13^2 = r^2 + 12^2 \] \[ 169 = r^2 + 144 \] \[ r^2 = 169 - 144 = 25 \] \[ r = \sqrt{25} = 5 \, \text{cm} \] Now, we can calculate the volume: \[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (5^2) (12) \] \[ V = \frac{1}{3} \times 3.14 \times 25 \times 12 = 314 \, \text{cm}^3 \] Converting to liters: \[ V = \frac{314}{1000} = 0.314 \, \text{liters} \]