Chapter 4: Problem 50
The domain of f^{\prime}\( is \)[0,1) \cup(1,2) \cup(2,3]
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Frictionless Cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time \(t=0\) to roll back and forth for 4 sec. Its position at time \(t\) is \(s=10 \cos \pi t .\) (a) What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? (b) Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then?
$$ \begin{array}{l}{\text { Analyzing Motion Data Priya's distance } D \text { in meters from a }} \\ {\text { motion detector is given by the data in Table 4.1. }}\end{array} $$ $$ \begin{array}{llll}{t(\text { sec) }} & {D(\mathrm{m})} & {t(\mathrm{sec})} & {D(\mathrm{m})} \\ \hline 0.0 & {3.36} & {4.5} & {3.59} \\ {0.5} & {2.61} & {5.0} & {4.15} \\ {1.0} & {1.86} & {5.5} & {3.99} \\ {1.5} & {1.27} & {6.0} & {3.37}\end{array} $$ $$ \begin{array}{llll}{2.0} & {0.91} & {6.5} & {2.58} \\ {2.5} & {1.14} & {7.0} & {1.93} \\ {3.0} & {1.69} & {7.5} & {1.25} \\ {3.5} & {2.37} & {8.0} & {0.67} \\ {4.0} & {3.01}\end{array} $$ $$ \begin{array}{l}{\text { (a) Estimate when Priya is moving toward the motion detector; }} \\ {\text { away from the motion detector. }} \\ {\text { (b) Writing to Learn Give an interpretation of any local }} \\ {\text { extreme values in terms of this problem situation. }}\end{array} $$ $$ \begin{array}{l}{\text { (c) Find a cubic regression equation } D=f(t) \text { for the data in }} \\ {\text { Table } 4.1 \text { and superimpose its graph on a scatter plot of the data. }} \\ {\text { (d) Use the model in (c) for } f \text { to find a formula for } f^{\prime} . \text { Use this }} \\ {\text { formula to estimate the answers to (a). }}\end{array} $$
Inscribing a Cone Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.
Wilson Lot Size Formula One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2}$$ where \(q\) is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be), \(k\) is the cost of placing an order (the same, no matter how often you order), \(c\) is the cost of one item (a constant), \(m\) is the number of items sold each week (a constant), and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). (a) Your job, as the inventory manager for your store, is to find the quantity that will minimize \(A(q) .\) What is it? (The formula you get for the answer is called the Wilson lot size formula.) (b) Shipping costs sometimes depend on order size. When they do, it is more realistic to replace \(k\) by \(k+b q,\) the sum of \(k\) and a constant multiple of \(q .\) What is the most economical quantity to order now?
Moving Shadow A man 6 ft tall walks at the rate of 5 \(\mathrm{ft} / \mathrm{sec}\) toward a streetlight that is 16 \(\mathrm{ft}\) above the ground. At what rate is the length of his shadow changing when he is 10 \(\mathrm{ft}\) from the base of the light?
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