ACT 1 Geo: Use length, with, parameter, circumference

$Q_{1}:$ In the following figure, the square $ABCD$ is a single side of a cube with a volume of $8$ cubic inches. In inches, what is the length of the line $BD $?

$Q_{2}:$ A rectangular container measuring $4$ feet wide, $8$ feet long, and $3$ feet tall is filled with a $1$-foot-deep layer of sand. In cubic feet, what volume of the container remains unfilled?

$Q_{3}:$ In inches, what is the radius of a sphere with a volume of $36\pi $?

$Q_{4}:$ When completely filled, a spherical balloon contains exactly $\frac{32\pi }{3}$ cubic feet of air. In feet, what is the radius of the balloon when it is completely filled?

$Q_{5}:$ In square meters, what is the area of a single face of a cube that has a volume of $125$ cubic meters?

$Q_{6}:$ A special garden design requires that the garden have three distinct square sections whose areas follow the ratio $2:3:5$. If such a garden is designed to have a total area of $1550$ square feet, then what would be the area of the smallest section in square feet?

$Q_{7}:$ The length of a rectangle is $40\% $ larger than its width. If the area of the rectangle is $140$ square feet, what is the width of the rectangle in feet?

$Q_{8}:$ In the following figure, $ABCD$ is a rectangle such that $\frac{x}{y} = \frac{1}{5}$. If the area of $ABPQ$ is $12$, what is the area of $ABCD $?

$Q_{9}:$ The length of the diagonal of a rectangle is $20$ feet. If the width of the rectangle is $12$ feet, what is its area in square feet?

$Q_{10}:$ The circles in the following figure are centered at point $O$. In square centimeters, what is the area of the shaded region?

$Q_{11}:$ In square units, what is the area of a circle with a circumference of $6\pi $?

$Q_{12}:$ In square meters, what is the area of a rectangle with a diagonal of $25$ meters and a width of $15$ meters?

$Q_{13}:$ Two squares of the same size overlap such that all three of the resulting rectangles shown in the figure have the same area. If the area of the shaded rectangle is $2$, what is the area of one of the original squares?

$Q_{14}:$ In the following figure, the circles centered at points $B$ and $D$ are tangent at the point $C$, and each circle has an area of $49\pi $. What is the length of the line segment $?

$Q_{15}:$ In the following figure, $FBCE$ is a rectangle, while $ABF$ and $ECD$ are congruent right triangles. What is the area of the quadrilateral $ABCD$?

$Q_{16}:$ Which of the following expressions represents the area of a right triangle with legs of lengths $x$ and $y $?

$Q_{17}:$ In the following figure, the area of the circle centered at the point $O$ is $\frac{29\pi }{4}$ square feet. Given that $AB$ has a length of $2$ feet, what is the area of triangle $ABC$ in square feet?

$Q_{18}:$ In the following figure, the square $ABCD$ is inscribed in the circle centered at point $O$. What is the area of the circle?

$Q_{19}:$ In square centimeters, what is the area of the parallelogram in the following figure?

$Q_{20}:$ A triangular prism is shown at below. Find the volume of the prism?

$Q_{21}:$ The figure shows an aluminum block of $10$in $\times 8$in $\times 12$in with an $8$in $\times 6$in $\times 12$in opening. What is the weight of the aluminum block to the nearest pound? (The density of aluminum is $0.098$ lb/ $in^{3}$)

$Q_{22}:$ The figure shows a pyramid with regular hexagonal base. The length of each side of the hexagonal face is $4$ units and the height of the pyramid is $7$ units. What is the volume of the pyramid?

$Q_{23}:$ A regular hexagonal prism with edge lengths of $2$ inches is created by cutting out a metal cylinder whose radius is $2$ inches and height is $4$ inches. What is the volume of the waste generated by creating the hexagonal prism from the cylinder, rounded to the nearest cubic inch?

$Q_{24}:$ In the figure shown, if all the water in the rectangular container is poured into the cylinder, the water level rises from $h$ inches to $(h+x)$ inches. Which of the following is the best approximation of the value of $x $?

$Q_{25}:$ In the figure, a double cone is inscribed in a cylinder whose radius is $x$ and height is $2x$ . What is the volume of the space inside the cylinder but outside the double cone, in terms of $x $?