7. let ( r ) be the triangular region in the first quadrant, with vertices at points ( (0,0) ), ( (0,2) )…

7. let ( r ) be the triangular region in the first quadrant, with vertices at points ( (0,0) ), ( (0,2) ), and ( (1,2) ). the region ( r ) is the base of a solid. for the solid, each cross section perpendicular to the ( y )-axis is an isosceles right triangle with the right angle on the ( y )-axis and one leg in the ( xy )-plane. what is the volume of the solid? (a) ( \frac{1}{3} ) (b) ( \frac{1}{2} ) (c) ( \frac{2}{3} ) (d) ( \frac{16}{3} )

7. let ( r ) be the triangular region in the first quadrant, with vertices at points ( (0,0) ), ( (0,2) ), and ( (1,2) ). the region ( r ) is the base of a solid. for the solid, each cross section perpendicular to the ( y )-axis is an isosceles right triangle with the right angle on the ( y )-axis and one leg in the ( xy )-plane. what is the volume of the solid? (a) ( \frac{1}{3} ) (b) ( \frac{1}{2} ) (c) ( \frac{2}{3} ) (d) ( \frac{16}{3} )

Answer

Explanation:

Step1: Find the equation of the line

The line passing through ((0,0)) and ((1,2)) has the equation (x=\frac{y}{2}) (using the slope - intercept form (y = mx + b), where (m = 2) and (b = 0), then solving for (x)).

Step2: Determine the area of the cross - section

Since each cross - section perpendicular to the (y) - axis is an isosceles right triangle with one leg (l=x) (in the (xy) - plane). The area of an isosceles right triangle (A=\frac{1}{2}l^{2}). Substituting (l = x=\frac{y}{2}), we get (A(y)=\frac{1}{2}(\frac{y}{2})^{2}=\frac{y^{2}}{8}).

Step3: Set up the volume integral

The limits of integration for (y) are from (y = 0) to (y = 2). Using the formula for the volume of a solid with known cross - sectional area (V=\int_{a}^{b}A(y)dy), we have (V=\int_{0}^{2}\frac{y^{2}}{8}dy).

Step4: Evaluate the integral

Using the power rule (\int y^{n}dy=\frac{y^{n + 1}}{n+1}+C(n\neq - 1)), we get (V=\frac{1}{8}\times\frac{y^{3}}{3}\big|_{0}^{2}). [ \begin{align*} V&=\frac{1}{24}(2^{3}-0^{3})\ &=\frac{8}{24}\ &=\frac{1}{3} \end{align*} ]

Answer:

A. (\frac{1}{3})