A cylindrical metal bar of diameter 14 cm and length 2m is melted and moulded into spherical balls.In the process,5% by volume is lost and what remains makes ball of radius 3.5cm.a)calculate the volume of metal used to make the balls.b)find to the nearest whole number the number of balls made.c)find the total surface area of the metal bar.find the total surface area of the balls made.Explain in detail.
Thanku
To solve this problem, we need to break it down into several steps. Let's go through each step one by one.
a) Calculate the volume of metal used to make the balls:
First, let's find the initial volume of the cylindrical metal bar before it is melted and molded into spherical balls. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given that the diameter of the bar is 14 cm, the radius (r) is half the diameter, so r = 14/2 = 7 cm = 0.07 m.
The height (h) of the bar is given as 2 m.
Using the formula, the initial volume of the metal bar is:
V_initial = π * (0.07^2) * 2
Now, we need to consider the loss of volume during the molding process. It is stated that 5% of the volume is lost.
So, the volume that remains after the loss can be calculated as:
V_remain = V_initial - (5/100) * V_initial
Now, we have the volume of the metal used to make the balls.
b) Find the number of balls made:
To calculate the number of balls, we need to divide the remaining volume of metal (V_remain) by the volume of each spherical ball.
The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius.
Given that the radius of each ball is 3.5 cm = 0.035 m.
Using the formula, the volume of each ball is:
V_each_ball = (4/3) * π * (0.035^3)
Now, we can find the number of balls made by dividing V_remain by V_each_ball.
c) Find the total surface area of the metal bar and the balls:
The total surface area of the cylindrical metal bar can be calculated using the formula for the lateral surface area of a cylinder: A = 2πrh.
Given that the height (h) is 2 m and the radius (r) is 0.07 m, we can calculate the total surface area of the bar.
To find the total surface area of the balls, we sum up the surface areas of all the individual balls produced. The formula for the surface area of a sphere is A = 4πr^2.
Given that the radius of each ball is 0.035 m, we can calculate the surface area of each ball and then multiply it by the number of balls obtained.
These calculations will give us the answers to parts (a), (b), and (c) of the question.
volume of original bar is πr^2 h = 9800π cm^3
after melting, 95% remains, or 9310π cm^3
(a) volume of each ball is 4/3 πr^3 = 57.17π cm^3
(b) # balls n = volume/57.17π
(c) area = 4πr^2 * n