In a baseball game, a batter hits a ground ball 26 feet in the direction of the pitcher's mound. See the figure below. The pitcher runs forward and reaches for the ball. At that moment, how far is the ball from first base? (Note: A baseball infield is a square that measures 90 feet on each side.)

Particles of mud are thrown from the rim of a rolling wheel. If the forward speed of the wheel is $v_{0}$, and the radius of the wheel is b, show that the greatest height above the ground that the mud can go is $b+\frac{v_{0}^{2}}{2 g}+\frac{g b^{2}}{2 v_{0}^{2}}$ At what point on the rolling wheel does this mud leave? (Note: It is necessary to assume that $v_{0}^{2} \geq b g$.)

A surveying team determines the height of a hill by placing a 12-foot pole at the top of the hill and measuring the angles of elevation to the bottom and the top of the pole. They find the angles of elevation to be as shown in the following figure. Find the height of the hill.

The minute hand of a clock is 1.2 centimeters long. How far does the tip of the minute hand travel in 40 minutes?

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. At what rate is his distance from third base increasing at the same moment?

A satellite weather map shows a hurricane off the coast of North Carolina. Use the information in the map to find the distance from the hurricane to Nags Head.

The following diagram shows two ways to play a golf hole. One is to hit the ball down the fairway on your first shot and then hit an approach shot to the green on your second shot. A second way is to hit directly toward the pin. Due to the water hazard, this is a riskier strategy. The distance $A B$ is 165 yards, $B C$ is 155 yards, and angle $A$ is $42.0^{\circ}$. Find the distance $A C$ from the tee directly to the pin. Assume that angle $B$ is an obtuse angle.

A regular hexagon is inscribed in a circle with a radius of exactly 40 centimeters. Find the exact length of one side of the hexagon.

The distance between an aircraft carrier and a Navy destroyer is 7620 feet. The angle of elevation from the destroyer to a helicopter is $77.2^{\circ}$, and the angle of elevation from the aircraft carrier to the helicopter is $59.0^{\circ}$. The helicopter is in the same vertical plane as the two ships, as shown in the following figure. Use these data to determine the distance $x$ from the helicopter to the aircraft carrier.

The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in length. The angle between the wing's leading edges $(\angle A B C)$ is $109.05^{\circ}$. What is the wing span (the distance from $A$ to $C$ ) of the B-2 Bomber?

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells. Find the relative growth rate.

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells.

Find the relative growth rate.

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P. The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

An airport runway is 3550 feet long and has an incline of $3.0^{\circ}$. The airport planning committee plans to replace this runway with a new runway, as shown in the following figure. The new runway will be inclined at an angle of $2.2^{\circ}$. What will be the length of the new runway?