Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 $ ^{\circ} F $ and 173 chirps per minute at 80 $ ^{\circ} F $.

(a) Find a linear equation that models the temperature $ T $ as a function of the number of chirps per minute $ N $. (b) What is the slope of the graph? What does it represent?(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

(a) Using $N$ in place of $x$ and $T$ in place of $y,$ we find the slope to be $\frac{T_{2}-T_{1}}{N_{2}-N_{1}}=\frac{80-70}{173-113}=\frac{10}{60}=\frac{1}{6} .$ So a linear\[\text { uation is } T-80=\frac{1}{6}(N-173) \Leftrightarrow \quad T-80=\frac{1}{6} N-\frac{173}{6} \quad \Leftrightarrow \quad T=\frac{1}{6} N+\frac{307}{6}\left[\frac{307}{6}=51.1 \overline{6}\right]\].(b) The slope of $\frac{1}{6}$ means that the temperature in Fahrenheit degrees increases one-sixth as rapidly as the number of cricketchirps per minute, Said differently, each increase of 6 cricket chirps per minute corresponds to an increase of $1^{\circ} \mathrm{F}$.(c) When $N=150,$ the temperature is given approximately by $T=\frac{1}{6}(150)+\frac{307}{6}=76.1 \overline{6}^{\circ} \mathrm{F} \approx 76^{\circ} \mathrm{F}$.

03:05

Jeffrey Payo

03:40

Heather Zimmers

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 2

Mathematical Models: A Catalog of Essential Functions

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Catherine A.

October 27, 2020

Heather Z., thanks this was super helpful.

This will help a lot with my midterm

Arkelly M.

September 10, 2021

how did find 307?

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

04:31

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

0:00

Biologists have noticed th…

07:01

05:35

05:39

09:56

06:35

17. Biologists have notice…

04:05

Biologists have observed t…

01:33

Crickets and Temperature. …

02:46

Crickets and Temperature B…

02:35

$$\begin{array}{l}{\te…

05:24

Entomologists have discove…

Right this is a co question on crickets, so we have information on the chirps per minute produced at different temperatures. Where t is the temperature in is chirps per minute. We have 2 pieces of data, but we know or were told that it's a linear relationship, and so our goal is to find a temperature as a function of number of turps. So that's kind of you could go outside and record chirps for in it and find the temperature cupica. Okay, we are used to finding lines of the form y equals m x, plus b, when we have a linear relationship. But in this case our main variable is going to be temperature in our independent variable x is going to be now in pot. So basically t has to equal to some slope which we're going to find times in plus a y intercept, so that will be our b, so we need to find both the slope and the y intercept. So, let's start with our data, we can find the slope slopes rise of which is like the y over the x, but for us that will be t over. So our slope and remember, slope is rise overrun which for us would be doing. Basically, it would go on top is undo that so we would have basically t 2 minus t 1 over in 2, minus n 1 point all right: let's use our data, so that will give us 80 minus 70 over 173 minus 113 point, and there are units. Just so we're aware, so the top is degrees f and the bottom is in terms per minute church per minute, all right. So those are the units okay, so we can clean this up a little bit. Our slope in is equal to 10 over 60 and that's in degrees f over i'm going to just put a c p m for chirps per minute, and we can reduce that all the way down to 1 sixpoint slope is degrees f and over trips per minute. That'S our slope, so that's perfect and that was a b b needed or question b had to do with slope. So let's go ahead and write that down the slope is 1 degree per churxe per minute. Okay. So that is our slope okay, but we want to come up with our equation, so let's keep going so right now so far we have t equal to 1 sixth of m and we still need to solve for unknown b. This is still our unknown. So, let's just use the top data point we'll plug in and solve for b intact. If i solve for b i could do or sorry let's go ahead and just substitute the end first there's more than 1 way to do it, but let's just substitute in okay. So we're going to go ahead and take our first play and substitute in so t is 113 and that's equal to 1 sixth of non. Oh, i plugged it in backwards. Let'S fix that real quick! We don't want to plug it in backwards. So t is 70 t is 70 and n is the 113 plus b it. So our goal is to solve for b. So we could subtract 113 over 6 from both sides and that will give us b. So this is 70 minus 113 over 6. So i'll read right on the right side b. Then, if i do common denominator, i can multiply 70 by 6 over 6, so 420, over 6 minus 113 over 6 and let's see what we get by calculator. So 420 minus 113 point. So if you leave it as a fraction b is 307 over 6 and if we go ahead, divide by 6, we'll get a decimal, basically 51.167. If we round to 3 decimal places, okay, so that's our b, so our part a then our t is equal to 1 sixth of n and i'll go ahead and leave it in exact form plus 307 over 6 point. So that is our temperature as a function of chirps permitted. All right, so we have done, is already done because the slope is the equal to 16 degrees fahrenheit for chirps per minute. So basically, every time we get an increase of 6 tips permitted, that's 1 degree. Fahrenheit increase in temperature, so critical. Okay. Now we want to use our relationship to basically approximate. What would the temperature be if we went outside we recorded 150 girfper minute and basically figured out the temperature? Now we can look in our data to see it's going to be somewhere in between 70 and 80 point. So that's good as a cut a reality check, but let's go ahead and use our equation. We can go ahead and we'll say approximately, because we're only going to approximate we're going to see if this is a a good model, but not perfect, 1, sixth of 150 chirps per minute and plus 307 over 6. So if we go 307 plus 150 on top, then our final answer is 457 over 6 and that is in degrees f. But we can go ahead and divide by 6 just to get a rough idea. And yes, it's good. If it's within that, between 70 and 80, as we expected so 76.2 degrees, f squish, that in there that is our separate, that that is our prediction for the temperature based on crickets chirping, so pretty cool all right. Hopefully, that helped have an amazing day.

View More Answers From This Book

Find Another Textbook

02:02

-6give the value of a32. You will get only three attempts on this proble…

03:30

Round 18.9999 to one decimal place.Select one O a 19.9b.18.9 0 c19.0…

02:01

2) Write the first five terms of an arithmetic sequence with a 5 and a commo…

Determine the equation of the 6th degree polynomial graphed below. Write you…

04:04

Juan buys peaches and grapefruit at the store. He writes the equations shown…

01:28

(10 pts) Use an iterated integral in spherical coordinates to represent the …

03:42

Atmospheric Pressure If the temperature is constant, then the atmospheric pr…

02:26

Resale Value Garland Mills purchased certain piece of machinery years ago fo…

02:44

Find the exact length of the curve_ X = 6 + 6t2, y = 2 + 4t3, 0 < t < …

Fill in the blank with "all" "no" or "some" …