LEAP 2025 Geometry Modeling & Application. Practice it free.

Set up the proportion $\dfrac{12\text{ kilograms}}{1\text{ cubic meter}} = \dfrac{540\text{ kilograms}}{x\text{ cubic meters}}\textsf{.}$ Cross-multiplying gives $12x = 540\textsf{,}$ so $x = 45\text{ m}^{3}\textsf{.}$

In a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, the short leg is half the hypotenuse: $\dfrac{16}{2} = 8\textsf{.}$

Both legs of a $45$-$45$-$90$ triangle are equal, so the other leg is $7$.

The center is $(9, 2)$ and the radius is $8\textsf{.}$ A point on the circle must satisfy $2 - 8 \le b \le 2 + 8\textsf{,}$ so $-6 \le b \le 10\textsf{.}$ The value $4$ is in this interval.

$y$-coordinate $= \dfrac{10 + 2}{2} = \dfrac{12}{2} = 6$.

Parallel lines have the same slope, so $m = \dfrac{2}{3}$. Substitute $(-3, 4)$ into $y = \dfrac{2}{3}x + b$: $4 = \dfrac{2}{3}(-3) + b = -2 + b$, so $b = 6$. The equation is $y = \dfrac{2}{3}x + 6\textsf{.}$

Parallel lines have the same slope. In $y = -x + 11$, the slope is $-1$, so the slope of line $b$ is also $-1\textsf{.}$

$V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3} \times 3 \times 9 \times 5 = 45$.