A person with a mass of $81 \mathrm{~kg}$ and a volume of $0.089 \mathrm{~m}^3$ floats quietly in water. If an upward force $F$ is applied to the person by a friend, the volume of the person above water increases by $0.0018 \mathrm{~m}^3$. Find the force $F$.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The volume of a pyramid is $\frac{1}{2}$ the volume of a rectangular solid having the same base and the same height. ____

The volume of a cube is 10$^6$ cubic feet. Simplify 10$^6$.

A cylinder with a moveable piston contains $0.87 \mathrm{~mol}$ of gas and has a volume of $334 \mathrm{~mL}$. What is its volume after an additional $0.22 \mathrm{~mol}$ of gas is added to the cylinder? (Assume constant temperature and pressure.)

One model for plant competition assume that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly dispersed over a wide area, the number of neighbors that a seedling may have usually follow a Poisson distribution with a mean equal to the density of seedlings per unit area. Suppose that the density of seedlings is four per square meter (m$^2$). a. What is the probability that a given seedling has no neighbors within 1 m$^2$? b. What is the probability that a seedling has at most three neighbors per m$^2$? c. What is the probability that a seedling has five or more neighbors per m$^2$? d. Use the fact that the mean and variance of a Poisson random variable are equal to find the proportion of neighbors that would fall into the interval $\mu\pm 2\sigma$. Comment on this result.

How can you minimize the surface area of an object with a fixed volume?

Approximate the volume of coating on a sphere of radius 4 in. if the coating is 0.02 in. thick.

Use the following data: Police radar on Interstate $80$ recorded the speeds of $110$ cars in a $65$ mi/h zone. The following table shows the class marks of the speeds recorded and the number of cars in each class.

Find the median.

The volume of a tetrahedron is known to be $\frac{1}{3}$ (area of base)(height). From this, show that the volume of the tetrahedron with edges $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ is $\frac{1}{6}|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|$.

a. Show that spheres occupy 74.0% of the total volume in a face-centered cubic structure in which all atoms are identical.

b. What percent of the total volume is occupied by spheres in a body-centered cube in which all atoms are identical?

An ideal monatomic gas, consisting of $2.5 \mathrm{~mol}$ of volume $0.086 \mathrm{~m}^3$, expands adiabatically. The initial and final temperatures are $25^{\circ} \mathrm{C}$ and $-68^{\circ} \mathrm{C}$. Calculate the final volume of the gas?

In 1787, Jacques Charles noticed that gases expand when heated and contract when cooled. A particular gas follows the model

$$y=\frac{5}{3} x+455,$$

where x is the temperature in Celsius and y is the volume in cubic centimeters. (Source: Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, MAA.)

(a) What is the volume when the temperature is $27^{\circ} \mathrm{C}$?

(b) What is the temperature when the volume is 605 cubic centimeters?

(c) Determine what temperature gives a volume of 0 cubic centimeters (that is, absolute zero, or the coldest possible temperature).

The volume of a sphere is given by $V=\frac{4}{3} \pi r^3$, where $r$ is the radius. Find the interval of values of the radius so that the volume is between $8 \ \mathrm{ft}^3$ and $12 \ \mathrm{ft}^3$, inclusive.

Find the volume of the cube prism having a side equal to 4 feet.