Math Problems and Some Notes

Suppose oil is leaking out a damaged tanker. The oil is forming a circular-shaped oil slick on the surface of the water. Suppose the radius t hours after the leak began is given by r(t) = 300t meters. Find a formula for the area (in terms of t) of the slick t hours after the leak began. How many hours has the oil been leaking if the slick is estimated to have an area of 10,000,000 square meters?

The number of bacteria in a certain culture at time t (in hours) is given by the formula Q(t) = 2(3^t), where Q9t) is measured in thousands. Find the number of bacteria at t=0, after 10 minutes, after 30 minutes, and after 1 hour.

Ten thousand dollars is invested in a savings account in which interest is compounded continuously at the rate of 11% per year. When will the account contain $35,000 How long will it take for money to double in the account?

A pond is stocked with 1,000 trout. Three months later, it is estimated that 600 remain. Find a formula of the form N(t) = Tb^t that can be used to estimate the number of trout remaining after t months.

A certain radioactive substance decays according to the formula q(t) = ae^(-0.0063t), where a is the initial amount of the substance and t is the time in days. Approximate the half-life of the substance.

Brielle and Elaine always travel to the Sunset Beach together for their summer vacation. Through the years, they return to the same section of beach and spend their time around the pier. Elaine has noticed that the water rises and falls on the posts of the pier and thinks the height of the water could be described as a sinusoidal function of time. Brielle was snorkeling and noticed that the pier posts are slimy up to the lowest level of the tide which was 2 feet for this particular post. Since Brielle and Elaine were on vacation, they had nothing better to do and found out there was a low point on the post at eight o’clock in the morning and a high point 40 inches higher at two o’clock in the afternoon. Find an equation of the height of the water relative to the post as a function of time from 12 o’clock midnight.
a. At 11:30 am, the girls wanted to take a walk on the beach but were afraid they might miss the high water mark so they stayed. What was the height of the water at 11:30 am?
b. Brielle’s little brother Sean is 2 feet 7 inches tall. When is the first time after 8 am he will be underwater if he plays right next to the post?

Lamaj rode his bike over a piece of gum. He continued riding his bike at a constant rate. At time t=1.25 seconds, the gum was at a maximum height above the ground and 1 second later the gum was at a minimum. If the wheel diameter is 68 cm, find a trigonometric equation that will find the height of the gum in cm at any time t.
a. Find the height of the gum when he gets to the corner at t=15.6 seconds if he maintains a constant speed.
b. Find the first and second time the gum reaches a height of 12 cm while he is riding at a constant rate.

Amanda was watching her little brother Mike play on a swing set. She decided that she would like to find his distance above the ground using a sine or cosine curve. She starts timing and finds that at t=2 s, Mike is at his highest point. He reaches his lowest point exactly 1.5 seconds later. Amanda also records that the highest Mike gets is 9 fee while the lowest point occurs at 1 foot. Write an equation that will find Mike’s height after t seconds.
a. Find Mike’s height at 5.4 seconds.
b. Find the first and second time that Mike reaches a height of 7.2 feet.

A pendulum hangs from a ceiling and swings back and forth towards a wall. Harry starts timing and at t=4 seconds the pendulum is closest to the wall, 25 cm away. Three seconds later the pendulum is farthest from the wall (83 cm). Find an equation for the distance the pendulum is from the wall at any time t.
a. Find out how far the pendulum is away from the wall at t=8 seconds.
b. Find the first time when the pendulum is 33 cm away from the wall.

A reflector on a bicycle wheel is 15 cm from the rim. The diameter of the wheel is 76 cm. At time = .5 seconds, the reflector is at its lowest point, .75 seconds later it returns to the same position. Find an equation which will locate the height of the reflector above the ground at any time t.
a. Find the height of the reflector when t=5.2 seconds.
b. Find the first time the reflector is at a height of 59 cm above the ground.

The temperature during the day can be approximated by a sinusoidal function. At 4 am, the temperature was at a low of 65 F. At 4 pm, the temperature hit a high of 103 F. Write an equation which will find the temperature t hours after midnight.
a. Find the temperature at 11 am.
b. Find the first time in the day when the temperature reaches 98 F.

The amount of air in a person’s lungs varies sinusoidally with time under normal breathing. When full, Karen’s lungs hold 2.8 liters of air. When empty, her lungs hold 0.6 liters of air. Her brother starts timing her breathing. At t=2 seconds, she has exhaled completely and at t=5 seconds, she had completely inhaled. Find a function that will find the amount of air in Karen’s lungs at any time.
a. Find the amount of air in Karen’s lungs if she starts holding her breath 3.5 seconds into the timing.
b. Find the first time Karen has 2.3 liters of air in her lungs.

The height of a piston in a cylinder can be modeled by a sine or cosine function. A piston is at its lowest point in a cylinder, 8 cm from the bottom, at t=3.2 seconds. The piston is at its highest position, 39 cm from the bottom, at t=3.6 seconds. Fund an equation for the height of the piston, in cm, at any given time t.
a. Find the height of the piston 15 seconds after the engine has started.
b. Find the first time the piston reaches 13 cm from the bottom.

Sean got a new yo-yo and noticed that the height of the yo-yo follows a sine or cosine curve. At time = 3 seconds the yo-yo is at its lowest height of 40 cm above the ground. The string is 62 cm long and one cycle takes 2 seconds. Find an equation that will determine the height of the yo-yo at any time t.
a. Find the height of the yo-yo after 20 seconds.
b. Find the first time the height of the yo-yo reaches 52 cm above the ground.

Cooper Toy Company has designed a new toy that uses a spring that follows a sinusoidal curve after you wind it up and start it. At t=5 seconds the end of the spring is at its highest point, 18 cm above the ground. Four seconds later, the spring is at its lowest point, which is 6 cm above the ground. Find an equation that will determine the height of the spring at any time t.
a. Find the height of the spring after 26 seconds.
b. Find the first time the height of the spring reaches 16 cm above the ground.

A plane flies 675 miles from A to B with a bearing of 75 degrees. Then it flies 140 miles from B to C with a bearing of 32 degrees. Find the angle at B between BA and due north. Find the measure of ABC. Find the straight line distance for the light from A to C. Find the bearing needed for a direct flight from A to C.

The current in a straight river is flowing so that if a motor boat travels at an angle of 20 degrees upstream with a velocity of 15km/hr, it will land at a spot on the opposite shore directly opposite its starting point. Find the speed of the current.

Find the magnitude and direction of the resultant two vectors: one with magnitude 50 and bearing 320 degrees and the other with magnitude 140 and bearing 120 degrees.

The famous World War I flying ace Red Baron is flying with a bearing of 155 degrees at a speed of 110 mph towards enemy lines when he encounters a strong wind of 60 mph blowing from 250 degrees. Draw and label a vector diagram to illustrate. Find the angel between the velocity vector of the Baron’s plane and the velocity vector of the wind. Find the Baron’s speed relative to the ground. Find the Baron’s actual bearing.

A plane flying at 230 mph with a bearing of 111 degrees encounters a wind blowing from a bearing of 263 degrees at 80 mph. draw and label a vector diagram for this situation. Find and label the angel between the plane’s velocity vector and the wind velocity vector. Find the plane’s true speed and bearing.

A channel flows from north to south at a rate of 15 mph. a sailboat heads at an angel of 30 degrees upstream at a rate of 40 mph. a strong wind blows towards the northwest at a rate of 30 mph. write each of the vectors in component form. Find the resultant vector in component form. What will be the craft’s true course and speed? If the channel is 20 miles wide, how long will it take to cross the channel? How far north from its starting position will the sailboat exit the channel?

A plane in calm air flies at a constant rate of 200 mph. There is a 40 mph wind blowing toward a bearing of 140 degrees. What bearing must the pilot fly in order to have a due east bearing? What rate will the plane be traveling?

A woman walks due west on the deck of an ocean liner at 2 mph. The ocean liner is moving due north at a speed of 25 mph. find the speed and direction of the woman relative to the surface of the water.

A ship sails for 50 miles on a bearing of 165 degrees then turns and sails on a bearing of 220 degrees for 35 more miles. How far is the ship from its starting point? What bearing would the ship need to take to head straight back to the start?

A ship sails for 25 miles on a bearing of 225 degrees then turns and sails on a bearing of 300 degrees for 16 more miles. How far is the ship from its starting point? At what bearing does the ship end up at with respect to where it started?

A function f is continuous at c if the following three conditions are met:
1. F(c) is defined.
2. Limx->c f(x) exists
3. Limx->c f(x) = f(c)