A solid right circular cone is 7 cm high, and the radius of its base is 22.2 cm. It is melted and recast into a right circular cone with radius of its base 3.7 cm. Then the height (in cm) of the cone is ______.
252
To find the height of the new cone, we start by understanding that when a solid object is melted and recast into a different shape, its volume remains unchanged. Let's calculate the volume of the original cone and set it equal to the volume of the newly formed cone to find the unknown height.
The volume of a right circular 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.
Step 1: Calculate the volume of the original cone
The original cone has a radius of \(22.2 \text{ cm}\) and a height of \(7 \text{ cm}\).
Volume of the original cone, \(V_{\text{original}} = \frac{1}{3} \pi (22.2)^2 \times 7\)
\(= \frac{1}{3} \pi \times 492.84 \times 7\)
\(= \frac{1}{3} \pi \times 3449.88\)
Step 2: Calculate the volume of the new cone with the given radius
The new cone has a radius of \(3.7 \text{ cm}\) and an unknown height \(h\).
Volume of the new cone, \(V_{\text{new}} = \frac{1}{3} \pi (3.7)^2 \times h\)
\(= \frac{1}{3} \pi \times 13.69 \times h\)
Since the volume is conserved, set the volumes equal:
\(\frac{1}{3} \pi \times 3449.88 = \frac{1}{3} \pi \times 13.69 \times h\)
We can cancel \(\frac{1}{3} \pi\) from both sides of the equation:
\(3449.88 = 13.69 \times h\)
Solve for \(h\):
\(h = \frac{3449.88}{13.69}\)
\(\approx 252 \text{ cm}\)
Thus, the height of the new cone is 252 cm.
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(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$
