Learning Curve A learning curve is a graph of a functionP(t) that measures the performance of someone learning askill as a function of the training time t. At first, the rate oflearning is rapid. Then, as performance increases andapproaches a maximal value M, the rate of learningdecreases. It has been found that the functionP(t) = M – Cewhere k and C are positive constants and C

Given: A learning curve is a graph of a function P(t) that measures the performance of someone…

Answered: Learning Curve A learning curve is a…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter4: Exponential And Logarithmic Functions

Section4.5: Exponential And Logarithmic Equations

Problem 102E

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Population Growth and Decline The graph shows the population P in a small industrial city from 1950 to 2000. Thevariable x represents the number of year since 1950. (a) Determine the intervals on which the function P isincreasing and on which it is decreasing. (b) What as the maximum population, and in what yearwas it attained? (c) Find the net change in the population P from 1970 to 1990.
Learning Curve A learning curve is a graph of a function P(t) that measures the performance of someone learning a skill as a function of the training time t. At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value M, the rate of learning decreases. It has been found that the function P(t)=MCekt where k and C are positive constants and CM is a reasonable model for learning. a Express the learning time t as a function of the performance level P. b For s pole-vaulter in training, the learning curve is given by P(t)=2014e0.024t where P(t) is the height he is able to pole-vault after t months. After how many months of training is he able to vault 12 ft? c Draw a graph of the learning curve in part b.
Sales Growth In this exercise, we develop a model for the growth rate G, in thousands of dollars per year, in sales of the product as a function of the sales level s, in thousands of dollars. The model assumes that there is a limit to the total amount of sales that can be attained. In this situation, we use the term unattained sales for difference this limit and the current sales level. For example, if we expect sales grow to 3 thousand dollars in the long run, then 3-s is the unattained sales. The model states that the growth rate G is proportional to the product of the sales level s, and the unattained sales. Assume that the constant of proportionality is 0.3 and that the sales grow to 2 thousand dollars in the long run. a.Find the formula for unattained sales. b.Write an equation that shows the proportionality relation for G. c.On the basis of the equation from the part b, make a graph of G as a function of s. d.At what sales level is the growth rate as large as possible? e.What is the largest possible growth rate?
Volume of a Box An open box is to constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each corner and folding up the sides, as shownin figure. (a) Express the volume V of the box as a function of x. (b) What is the domain of V? (Use the fact that length and volume must be positive.) (c) Draw a graph of the function V, and use it to estimate the maximum volume for such a box.
Fluid Flow The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine whether the input pipe and each drainpipe are open or closed in specific subintervals of the 1 hour of time shown in the graph. (There are many correct answers.)
Arboriculture The growth of a red oak tree is approximated by the function G=0.003t3+0.137t2+0.458t0.839,2t34 where G is the height of the tree (in feet) and t is its age (in years). (a) Use a graphing utility to graph the function. (b) Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year. (c) Using calculus, the point of diminishing returns can be found by finding the vertex of the parabola y=0.009t2+0.274t+0.458. Find the vertex of this parabola. (d) Compare your results from parts (b) and (c).
Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. Suppose g(x)=f(x) where f(x)0 for all x. Explain why the local minima and maxima of f and g occur at the same values of x. Let gx be the distance between the point 3,0 and the point (x,x2) on the graph of the parabola y=x2. Express g as a function of x. Find the minimum value of the function g that you found in part b. Use the principle described in part a to simplify your work.
Changing Water Levels: The graph shows the depth of water Win a reservoir over a one-year period as a function of the number of days xsince the beginning of the year. (a) Determine the intervals on which the function W is increasing and on which it is decreasing. (b) At what value of x does W achieve a local maximum? A local minimum? (c) Find the net change in the depth W from 100 days to 300 days
Population Growth The projected population of the United States for the years 2025 through 2055 can be modeled by P=307.58e0.0052t, where P is the population (in millions) and t is the time (in years), with t=25 corresponding to 2025. (a) Use a graphing utility to graph the function for the years 2025 through 2055. (b) Use the table feature of the graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, during what year will the population of the United States exceed 430 million?

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