Q 44 Find each mass described in Exer... [FREE SOLUTION]

Chapter 6: Q 44 (page 556)

Find each mass described in Exercises 41鈥�48.
The mass of a cylindrical rod with a radius of 3 centimeters and a length of 20 centimeters, made of a combination of copper and aluminum in such a way that
the density of the rod x centimeters from the left end is $蚁 (x) = 8.93 鈭� 0.015 x$ grams per cubic centimeter

Short Answer

Expert verified

The mass of the rod is 5,134.62 grams

Step by step solution

Given Information

The mass of a cylindrical rod with a radius of 3 centimeters and a length of 20 centimeters . the density of the rod x centimeters is $蚁 (x) = 8.93 鈭� 0.015 x$

Formula

Volume of cylinder is ${蟺谤}^{2} h$

where 'r' is the radius and h is the height of the cylinder

Each slice is of radius 3 and width $鈭� x$

So, the volume becomes

${蟺谤}^{2} h 蟺 {(3)}^{2} 鈭� x 9 蟺鈭� x$

Density

Density of entire slice is

$蚁 (x_{i}) = 8.93 鈭� 0.015 x_{i} 蚁 = \frac{M}{V} 8.93 鈭� 0.015 x_{i} = \frac{M}{9 蟺鈭� x} M = (8.93 鈭� 0.015 x_{i}) 9 蟺鈭� x$

Sum

approximate mass is given by sum

$\underset{i}{鈭� (8.93 鈭� 0.015 x_{i}) 9 蟺鈭� x} To get total mass, accumulate the mass from x = 0 to 20 Mass = {鈭�}_{0}^{20} (8.93 鈭� 0.015 x) 9 蟺诲虫 = 9 蟺 {鈭�}_{0}^{20} (8.93 鈭� 0.015 x) dx = 9 蟺 {[8.93 x - \frac{0.015 x^{2}}{2}]}_{0}^{20} = 9 蟺 [8.93 (20) - \frac{0.015 {(20)}^{2}}{2}] = 5134.62 grams$