Q. 45 - Calculus

Question

Find each mass described in Exercises 41–48.

The mass of a 10-centimeter rod with square cross section of side length 1 centimeter, made of a combination of metals in such a way that the density of the rod x centimeters from the left end is $\rho \left(x\right) = 15 - 0.5x$ grams per cubic centimeter.

Step-by-Step Solution

Answer

The mass of the rod is 125 grams

Step 1 : Given Information

The mass of a 10-centimeter rod with square cross section of side length 1 centimeter,  with the density of the rod x centimeters   is  

$\rho \left(x\right) = 15 - 0.5x$

Step 2: Fromula

Volume of cuboid = length time width times height

The length and height of cuboid is 1 inches  and width is $∆x$

$$\begin{gathered} Volume =lbh \\ V=\left(1\right)\left(1\right)∆x \\ V=∆x \end{gathered}$$

Step 3: Density

Let xi represents slice of the rod at some point 

The density throughout enter slice is 

$$\begin{gathered} \rho \left(x_i\right) = 15 - 0.5x_i \\ \rho =\frac{M}{V} \\ 15 - 0.5x_i=\frac{M}{∆x} \\ M=∆x\left( 15 - 0.5x_i\right) \end{gathered}$$

Step 4: Integration

The approximate of the entire rod is the sum  

$\sum _{}∆x\left( 15 - 0.5x_i\right)$

Accumulate the mass from x= 0 to 10 to find total mass

$$\begin{gathered} {\int }_0^{10}\left( 15 - 0.5x\right)dx \\ ={\left[15x-\frac{0.5x^2}{2}\right]}_0^{10} \\ =15\left(10\right)-\frac{0.5{\left(10\right)}^2}{2} \\ =125 grams \end{gathered}$$