Learning Materials
Features
Discover
Chapter 1: Problem 9
Find the slope of the line through \(P\) and \(Q .\) \(P(-1,2), Q(0,0)\)
Over 30 million students worldwide already upgrade their learning with 91影视!
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for free
Fish Population The fish population in a certain lake rises and falls according to the formula $$F=1000\left(30+17 t-t^{2}\right)$$ Here \(F\) is the number of fish at time \(t,\) where \(t\) is measured in years since January \(1,2002,\) when the fish population was first estimated. (a) On what date will the fish population again be the same as it was on January \(1,2002 ?\) (b) By what date will all the fish in the lake have died?
Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.
Shrinkage in Concrete Beams As concrete dries, it shrinks - the higher the water content, the greater the shrinkage. If a concrete beam has a water content of \(\bar{w} \mathrm{kg} / \mathrm{m}^{3},\) then it will shrink by a factor $$S=\frac{0.032 w-2.5}{10,000}$$ where \(S\) is the fraction of the original beam length that disappears due to shrinkage. (a) A beam \(12.025 \mathrm{m}\) long is cast in concrete that contains \(250 \mathrm{kg} / \mathrm{m}^{3}\) water. What is the shrinkage factor \(S ?\) How Iong will the beam be when it has dried? (b) A beam is \(10.014 \mathrm{m}\) long when wet. We want it to shrink to \(10.009 \mathrm{m},\) so the shrinkage factor should be \(S=0.00050 .\) What water content will provide this amount of shrinkage? PICTURE CANT COPY
If \(a_{1}, a_{2}, \ldots, a_{n}\) are nonnegative numbers, then their arithmetic mean is \(\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\) and their geometric mean is \(\sqrt[n]{a_{1} a_{2} \dots a_{n}}\). The arithmetic-geometric mean inequality states that the geometric mean is always less than or equal to the arithmetic mean. In this problem we prove this in the case of two numbers \(x\) and \(y .\) (a) If \(x\) and \(y\) are nonnegative and \(x \leq y,\) then \(x^{2} \leq y^{2}\) [ Hint: First use Rule 3 of Inequalities to show that \(\left.x^{2} \leq x y \text { and } x y \leq y^{2} .\right\\}\) (b) Prove the arithmetic-geometric mean inequality $$\sqrt{x y} \leq \frac{x+y}{2}$$
It follows from Kepler's Third Law of planetary motion that the average distance from a planet to the sun (in meters) is $$d=\left(\frac{G M}{4 \pi^{2}}\right)^{1 / 3} T^{2 / 3}$$ where \(M=1.99 \times 10^{30} \mathrm{kg}\) is the mass of the sun, \(G=6.67 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}\) is the gravitational constant, and \(T\) is the period of the planet's orbit (in seconds). Use the fact that the period of the earth's orbit is about 365.25 days to find the distance from the earth to the sun.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine