Modeling Using Function Notation Answers

A ball is being inflated at a rate that can be modeled by the function V(t) = 10t, where t is the number of seconds spent inflating the ball. The radius at a given volume can be modeled by the function .Which composite function can be used to determine the radius of the ball depending on the number of seconds spent inflating the ball?

A profit formula for dining plates from the previous year was modeled by the function P(d) = –15d2 + 1,200d – 2,000. The manufacturer noticed an increase in the number of units sold by 50% the next year, which can be modeled by the function l(d) = 1.5d.

The function can be used to determine the side length, x, of a cube given the surface area of the cube, A. The function V(x) = x3 can be used to find the volume of a cube. How can these functions be combined?

One brand of coffee is packaged in cylinders where the height is equal to the radius, r. The volume of the package, in cubic centimeters, can be found using the function V(r) = πr3. The number of ounces of coffee in the cylinder depends on the volume of the cylinder, V, in cubic centimeters. This can be modeled by the function C(V) = 3.2V.

The density of titanium is 4.5 g/cm3. A titanium object is accelerating at a rate of 800 cm/s2. The mass of the object can be modeled by the function m(v) = 4.5v, where v is the volume in cubic centimeters. Additionally, the force of the object can be found using the function F(m) = 800m.

A roller coaster starts with the cars being pulled up a ramp. The mass of the cars is estimated by the function m(p) = 175p + 1,180 where p is the number of passengers in the car. The potential energy of the car is calculated using the function E(m) = 9.8mh, where h is the height of the ramp.

A business plan is centered around a circular area of distribution. The plan calls for the radius of the distribution to increase 50 miles per year. The investors have asked that the plan include a reference to the area of distribution based on the number of years in business. These functions model this plan:Radius in y years: r(y) = 50y Total area: A(r) = πr2; r in miles

A flare is designed to follow the path modeled by the function h(t) = –16t2 + 100t, where t is the time elapsed, in seconds, and h(t) is the height, in feet, of the flare at that time. The function can be used to convert the height in feet to the height in meters.

An airplane’s average speed from San Diego to Dallas can be calculated using the function where t is the number of hours of travel. When there are tailwinds, the increased speed can be calculated using the functionw(s) = fs, where f is the percent increase as a decimal. On a given day, the tailwinds caused the average speed to increase by 2.5%, or w(s) = 1.025s.

The function r(x) = –0.000345(x – 50)2 + 2.1 can be used to model the curve of a road designed to allow for water to run off in rainstorms depending on the distance from the shoulder of the road, x, in feet. A European firm studying these curves is going to convert feet to meters using the function m(x) = 0.3048x.