Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Bores Under certain conditions, tsunami waves encountering land will develop into bores. A bore is a surge of water much like what would be expected if a dam failed suddenly and emptied a reservoir in to a river bed. In the case of a bore travelling from the ocean into a dry river bed, one study shows that the velocity
where
a. A bore travels up a dry river bed. How does the velocity of the tip compare with its initial velocity when its height is reduced to half of its initial height?
b. How does the height of the bore compare with its initial height when the velocity of the tip is reduced to half of its initial velocity?
c. If the tip of one bore surging up a dry river bed is three times the height of another, how do their velocities compare?
Want to see the full answer?
Check out a sample textbook solutionChapter 5 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. The Rock with a Changed Reference Point Make graphs of position and velocity for a rock tossed upward from ground level as it might be viewed by someone standing atop a tall building. Thus, the location of the rock is measured by its distance down from the top of the building.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Giants Ants and Spiders Many science fiction movies feature animals such as ants, spiders, or apes growing to monstrous sizes and threatening defenseless Earthlings. Of course, they are in the end defeated by the hero and heroine. biologists use power function as a rough guide to relate body weight and cross-sectional area of limbs to length or height. Generally, weight is thought to be proportional to the cube of length, whereas the cross-sectional area of limbs is proportional to the square of length. Suppose an ant, having been exposed radiation is enlarged to 500 times its normal length. Such an event can occur only in Hollywood fantasy. Radiation is utterly incapable of causing such a reaction. a.By how much will its weight be increased? b.By how much will the cross-sectional area of its legs be increased? c.Pressure on a limb is weight divided by cross-sectional area. By how much has the pressure on a leg of the giant ant increased? What do you think is likely to happen to this unfortunate ant? Note: The factor by which pressure increases is given by . FactorofincreaseinweightFactorofincreaseinarea)arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Hair Growth When you are 18 years old you have a hair that is 14 centimeters long, and your hair grows about 12 centimeters each year. Let H(t) be the length, in centimeters, of that hair t years after age 18. a. Find a formula that gives H as a linear function of t. b. How long will it take for the hair to reach a length of 90 centimeters?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. A Rubber Ball A rubber ball is dropped from the top of a building. The ball lands on concrete and bounces once before coming to rest on the grass. Measure the location of the ball as its distance up from the ground. Make graphs of the location and velocity of the ball.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Looking Up The constant g32feetpersecondpersecond is the downward acceleration due to gravity near the surface of the Earth. If we stand on the surface of the Earth and locate objects using their distance up from the ground, then the positive direction is up, so down is the negative direction. With this perspective, the equation of change in velocity for a freely falling object would be expressed as dVdt=g. We measure upward velocity V in feet per second and time t in seconds. Consider a rock tossed upward from the surface of the Earth with an initial velocity of 40feetpersecond upward. a. Use a formula to express the velocity VV(t) as a linear function. Hint: You get the slope of V from the equation of change. The vertical intercept is the initial value. b. How many seconds after the toss does the rock reach the peak of its flight? Hint: What is the velocity of the rock when it reaches its peak? c. How many seconds after the toss does the rock strike the ground? Hint: How aces me time it takes for the rock to rise to its peak compare with the time it takes for it to fall hack to the ground?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Nail Growth The rate of fingernail growth depends on many factors, but in adults, nails grow at an average rate of 3 millimeters per month. If a nail is initially 12 millimeters long, find a formula that gives the length L, in millimeters, of the nail if left unclipped after t months.arrow_forward
- ReminderRound all answers to two decimal places unless otherwise indicated. DensityThe total weight of a rock depends on its size and is proportional to its density. In this context, density is the weight per cubic inch. Let w denote the weight of the rock in pounds, s the size of the rock in cubic inches, and d the density of the rock in pounds per cubic inch. a. What is the total weight of a 3-cubic-inch rock that weighs 2 pounds per cubic inch? b. Write an equation that shows the proportionality relation. What is the constant of proportionality? c. Use the equation you found in part b to find the total weight of a 14-cubic-inch rock with density 0.3 pound per cubic inch.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Arterial Blood Flow Medical evidence shows that a small change in the radius of an artery can indicate a large change in blood flow. For example, if one artery has a radius only 5 larger than another, the blood flow rate is 1.22 times as large. Further information is given in the table below. Increase in radius Times greater blood flow rate 5 1.22 10 1.46 15 1.75 20 2.07 a. Use the average rate of change to estimate how many times greater the blood flow rate is in an artery that has a radius 12 larger than another. b. Explain why if the radius is increased by 12 and then we increase the radius of the new artery by 12 again, the total increase in the radius is 25.44. c. Use parts a and b to answer the following question: How many times greater is the blood flow rate in an artery that 25.44 larger in radius than another? d. Answer the question in part c using the average rate of change.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Height of Tsunami WavesWhen waves generated by tsunamis approach shore, the height of the waves generally increases. Understanding the factors that contribute to this increase can aid in controlling potential damage to areas at risk. Greens law tells how water depth affects the height of a tsunami wave. If a tsunami wave has height H at an ocean depth D, and the wave travels to a location with water depth d, then the new height h of the wave is given by h=HR0.25, where R is the water depth ratio given by R=D/d. a. Calculate the height of a tsunami wave in water 25feet deep if its height is 3feet at its point of origin in water 15,000feet deep. b. If water depth decreases by half, the depth ratio R is doubled. How is the height of the tsunami wave affected?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Decibels Sound exerts a pressure P on the human ear. This pressure increases as the loudness of the sound increases. It is convenient to measure the loudness D in decibels and the pressure P in dynes per square centimeter. It has been found that each increase of 1decibel in loudness causes a 12.2 increase in pressure. Furthermore, a sound of loudness 97decibels produces a pressure of 15 dynes per square centimeter. a. Explain why P is an exponential function of D and find the growth factor. b. Find P(0) and explain in practical terms what your answer means. c. Find an exponential model for P as a function of D. d. When pressure on the ear reaches a level of about 200 dynes per square centimeter, physical damage can occur. What decibel level should be considered dangerous?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Length of Skid Marks Versus Speed When a car skids to a stop, the length L, in feet, of the skid marks is related to the speed S, in miles per hour, of the car by the power function L=130hS2. Here the constant h is the friction coefficient, which depends on the road surface. For dry concrete pavement, the value of h is about 0.85. a. If a driver going 55milesperhour on dry concrete jams on the brakes and skids to a stop. how long will the skid marks be? b. A policeman investigating an accident on dry concrete pavement finds skid marks 230feet long. The speed limit in the area is 60milesperhour. Is the driver in danger of getting a speeding ticket? c This part of the problem applies to any road surface, so the value of h is not known. Suppose you are driving at 60milesperhour, but, because of approaching darkness, you wish to slow to a speed that will cut your emergency stopping distance in half. What should your new speed be? Hint: You should use the homogeneity property of power functions here. By what factor should you change your speed to ensure that L changes by a factor of 0.5?arrow_forwardReminder:-Round all answers to two decimal places unless otherwise indicated. Head and Pressure Determining the water pressure at a given location employs the concept of the head, which is the vertical distance, in feet, from the surface of a source body of water to the location. The pressure exerted by water is proportional to the head. If we measure head in feet and pressure in pounds per square inch, then the constant of proportionality is the weight of a column of water that is 1 foot high and 1 inch square at the base. That much water weighs 0.434 pound See figure 1.57. FIGURE 1.57 a. Write an equation that expresses the proportionality relationship between pressure p and head h. b. For a pumper truck pumping water to a fire, the back pressure is the additional pressure on the pump caused by the height of the nozzle. Consider a pumper at street level pumping water through a hose to firefighters on the top of the eighth floor of a building. If each floor is 12 feet high, what is the head of water at the mouth of the nozzle? What is the back pressure on the pumper? Another way of thinking of back pressure is as the minimum pressure the pumper must produce in order to make water flow out the end of the nozzle. c. Head and therefore back pressure depends only on the height of the nozzle above the pumper. It is affected neither by the volume of the water nor by horizontal distance. A pumper in a remote location is pumping water to firefighters on the far slope of a hill. At its peak, the hill is 185 feet higher than the pumper. The hose goes over the hill and then down the hill to a point 40 feet below the peak. Find the head and the back pressure on the pumper.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning