A cylinder 15 cm long and 9 cm in radius is
made of two different metals bonded end-to-
end to make a single bar. The densities are
4.3 g/cm3 and 6.3 g/cm3.
15 cm
9 cm radius
What length of the lighter-density part of
the bar is needed if the total mass is 21757 g?
Answer in units of cm.
0
metal A length x
metal B length (15-x)
x(4.3)(pi r^2) + (15-x)(6.3)(pi r^2)
= 21757
4.3 x + 94.3 - 6.3 x = 21757/(81 pi)
2 x = 8.8
x = 4.4 cm
To find the length of the lighter-density part of the bar, we need to set up an equation based on the given information.
Let's denote the length of the lighter-density part of the bar as "x" (in cm). Therefore, the length of the heavier-density part of the bar will be "15 - x" (since the total length of the cylinder is 15 cm).
The volume of a cylinder is given by the formula: V = πr^2h, where "r" is the radius and "h" is the height (or length) of the cylinder.
Using the given information, we can calculate the volume of the lighter-density part and the heavier-density part separately.
For the lighter-density part:
Volume = π(9^2) * x = 81πx cm^3
For the heavier-density part:
Volume = π(9^2) * (15 - x) = 81π(15 -x) cm^3
Since density is defined as mass divided by volume, we can write the following equation based on the given information:
(4.3 g/cm3)(81πx) + (6.3 g/cm3)(81π(15 - x)) = 21757 g
To solve this equation, we'll follow these steps:
1. Distribute the multiplication:
(4.3 g/cm3)(81πx) + (6.3 g/cm3)(81π(15)) - (6.3 g/cm3)(81πx) = 21757 g
2. Simplify further:
347.7πx + 12258π - 510.3πx = 21757 g
3. Combine like terms:
-162.6πx + 12258π = 21757 g
4. Move the terms containing "x" to one side of the equation:
-162.6πx = 21757 g - 12258π
5. Divide both sides of the equation by -162.6π:
x = (21757 g - 12258π) / (-162.6π)
Now we can calculate the value of x using a calculator:
x ≈ 14.966 cm
Therefore, the length of the lighter-density part of the bar (x) is approximately 14.966 cm.