Q 46 - Calculus

Question

Find each mass described in Exercises 41–48.

The mass of a rectangular block of metal that is 30 centimeters wide, 21 centimeters long, and 14 centimeters tall, given that its density y centimeters from its base is

$\rho ( y) = 6.8 + 0.14y$ grams per cubic centimeter.

Step-by-Step Solution

Answer

The mass of the rod is 78,498 grams

Step 1: Given Information

 The mass of a rectangular block of metal that is 30 centimeters wide, 21 centimeters long, and 14 centimeters tall . its density y centimeters from its base is 

$\rho ( y) = 6.8 + 0.14y$

Step 2: volume

Volume of cuboid = length times width times height

The length and height of cuboid is 21 cm  and height is 14 cm. The width is $∆y$

 $$\begin{gathered} Volume = \left(21\right)\left(14\right)∆y \\ V= 294∆y \end{gathered}$$

The density of entire slice is   ${\rho }_k( y) = 6.8 + 0.14y_k$

Density formula is 

$$\begin{gathered} \rho =\frac{M}{V} \\ 6.8 + 0.14y_k=\frac{M}{294∆y} \\ M=\left( 6.8 + 0.14y_k\right)294∆y \end{gathered}$$

Step 3: Mass

Approximate  mass of the entire rod is the sum 

$\sum _{}\left( 6.8 + 0.14y_k\right)294∆y$

Accumulate the mass from y=0 to y=10  to find total mass of the rod 

$$\begin{gathered} {\int }_0^{10}\left( 6.8 + 0.14y\right)294∆y \\ =294{\int }_0^{30}\left( 6.8 + 0.14y\right)∆y \\ =294{(6.8y+0.07y^2)}_0^{30} \\ =294(6.8(30)+0.07{\left(30\right)}^2) \\ = 78498 grams \end{gathered}$$