Modeling Data The circumference C (in inches) of a vase is measured at three-inch intervals starting at its base. The measurements are shown in the table, where y is the vertical distance in inches from the base.
y |
0 |
3 |
6 |
9 |
12 |
15 |
18 |
C |
50 |
65.5 |
70 |
66 |
58 |
51 |
48 |
(a) Use the data to approximate the volume of the vase by summing the volumes of approximating disks.
(b) Use the data to approximate the outside surface area (excluding the base) of the vase by summing the outside surface areas of approximating frustums of right circular cones.
(c) Use the regression capabilities of a graphing utility to find a cubic model for the points (y, r), where
(d) Use the model in part (c) and the integration capabilities of a graphing utility to approximate the volume and outside surface area of the vase. Compare the results with your answers in parts (a) and (b).
Trending nowThis is a popular solution!
Chapter 7 Solutions
Calculus
- Projectile Motion In Exercises 75 and 76, consider the path of an object projected horizontally with a velocity of v feet per second at a height of s feet, where the model for the path is x2=v216ys. In this model (in which air resistance is disregarded), y is the height (in feet) of the projectile and x is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 100-foot tower with a velocity of 28 feet per second. (a) Write an equation for the parabolic path. (b) How far does the ball travel horizontally before it strikes the ground?arrow_forwardTEST YOUR UNDERSTANDING Suppose we start with 9 milligrams of pollutant per liter of water, as in the example. Assume that the cleaning process decreases the concentration of pollutant by 30 each hour. What is the pollutant concentration after each of the first 3 hours? EXAMPLE 1.9 PURIFYING WATER Water that is initially contaminated with a concentration of 9 milligrams of pollutant per liter of water is subjected to a cleaning process. The cleaning process is able to reduce the pollutant concentration by 25 each hour. Let C=C(t) denote the concentration, in milligrams per liter, of pollutant in the water r hours after the purification process begins. Part 1 What is the concentration of pollutant in the water after 3 hours? Part2 Compare the results predicted by the verbal description at the end of each of the first 3 hours with those given by the formula C=90.75t. Part 3 The formula given in part 2 is, in fact, the correct one. Use it to find the concentration of pollutant after 414 hours of cleaning.arrow_forwardMeasurement Use the fact that 14 gallons is approximately the same amount as 53 liters to find a mathematical model that relates liters y to gallons x. Then use the model to find the numbers of liters in 5 gallons and 25 gallons.arrow_forward
- Finding volumearrow_forwardA public school district furnishes pencils to its elementary students. The pencils the secretary orders from the district warehouse each year are packaged in boxes that contain one gross (12 dozen=1 gross) pencils. The average use of pencils is 9.3 pencils per student over the school year. Of the school’s enrollment is 812 students, what is minimum number of boxes of pencils that should be ordered? Use dimensional analysis.arrow_forwardMATHEMATICAL CONNECTIONS: In Exercises 41–42, point D is the centroid of △ABC. Use the given information to find the value of x. Attach your answer in image filearrow_forward
- Calculus: Use a local LINEAR approximation to estimate 9.021.5.arrow_forwardHeckscher-Ohlin model of trade • How is the relative world price determined after trade?• How do you demonstrate gains from trade?• What are the key theorems/principles that result from the model?arrow_forwardMinimum painting surface A metal cistern in the shape ofa right circular cylinder with volume V = 50 m3 needs to bepainted each year to reduce corrosion. The paint is applied onlyto surfaces exposed to the elements (the outside cylinder wall andthe circular top). Find the dimensions r and h of the cylinder thatminimize the area of the painted surfaces.arrow_forward
- Calculate the volume and surface area of a standard cereal box. Calculate the cost of manufacturing a standard cereal box if cardboard costs $0.05 per square inch. Design a new cereal box that will hold the same amount of cereal but reduce manufacturing costs. Prove that your new design holds the same amount but can be manufactured more cheaply. Draw a net of your new design. Include all relevant measurements so that your cereal box can be mass-produced. a table containing the measurements of a standard cereal box clearly labelled work showing how you calculated the volume of a standard cereal box clearly labelled work showing how you calculated the surface area of a standard cereal box clearly labelled work showing how you found the cost of producing a standard cereal box an explanation proving that your new design holds the same amount of cereal as a standard cereal box but that it costs less to manufacture a net of your new design, including any measurements that would be…arrow_forwardA rancher plan to fence a rectangular fence adjacent to a river. The rancher has 100m of fencing and no fence along the river on one side. Find the dimension that yield the maximum amount area for the pasture.arrow_forwardCalculus 2 Having trouble with how to find the area of the region enclosed by the curves (blue highlighted area). Please help. Thank you, Braidenarrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage