How many silver coins ,2cm in diameter and of thickness 2mm,must be melted to form a showpiece in the form of frustum of a cone having height 9cm and radii of outer circular ends as5cm 3cm
I assume you can find the volume of a cylindrical coin.
For the frustrum, consider it as a large cone with a small cone cut off the top. If you draw a side view, then if the height of the small top cone is h, you have
(h+9)/5 = h/3
h = 27/2
Now you see that the volume of the frustrum is
π/3 (5^2*(9+27/2) - 3^2*27/2) = 147π
Now just divide the volume of the frustrum by that of the coin to get the number of coins.
To find the number of silver coins required to form a showpiece in the form of a frustum of a cone, we need to calculate the volume of the frustum and then divide it by the volume of a single coin.
Here's how you can do it step by step:
Step 1: Calculate the volume of the frustum of the cone.
The formula for the volume of a frustum of a cone is:
V = (1/3) * π * h * (r₁² + r₂² + r₁ * r₂)
Given:
h = 9 cm (height of the frustum)
r₁ = 5 cm (radius of the larger circular end)
r₂ = 3 cm (radius of the smaller circular end)
Using the formula, we can plug in the values:
V = (1/3) * π * 9 * (5² + 3² + 5 * 3)
V = (1/3) * π * 9 * (25 + 9 + 15)
V = (1/3) * π * 9 * 49
V = 147π cm³
Step 2: Calculate the volume of a single silver coin.
The volume of a coin can be calculated using the formula for the volume of a cylinder:
V_coin = π * r² * h_coin
Given:
r_coin = 1 cm (radius of the coin, which is half the diameter)
h_coin = 0.2 cm (thickness of the coin, which is the same as the given thickness in mm)
Using the formula, we can plug in the values:
V_coin = π * (1²) * 0.2
V_coin = π * 0.2
V_coin = 0.2π cm³
Step 3: Divide the volume of the frustum by the volume of a single coin to find the number of coins required.
Number of coins = V_frustum / V_coin
Number of coins = 147π / 0.2π
Number of coins = 147 / 0.2
Number of coins = 735 coins
Therefore, you would need 735 silver coins, each 2 cm in diameter and 2 mm in thickness, to form a showpiece in the form of a frustum of a cone with a height of 9 cm and outer radii of the circular ends as 5 cm and 3 cm.