Download the App!

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

Question

Answered step-by-step

A roast turkey is taken from an oven when its temperature has reached $ 185^o F $ and is placed on a table in a room where the temperature is $ 75^o F. $(a) If the temperature of the turkey is $ 150^o F $ after half anhour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to $ 100^o F? $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

03:49

Wen Zheng

01:49

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 8

Exponential Growth and Decay

Derivatives

Differentiation

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

06:29

A roast turkey is taken fr…

01:27

02:49

A roasted turkey is taken …

03:39

06:45

08:19

03:38

11:27

A roast turkey is removed …

04:23

A turkey is pulled from th…

06:02

Use Newton's Law of C…

05:05

You are cooling a turkey t…

09:44

These exercises use Newton…

this problem uses Newton's law of cooling, and we see when we read through the lesson in the book that Newton's law of cooling fits the model for exponential growth and decay y equals. Why not eat to the Katie? And we can also make it a little bit more problems specific if we replace the why with T minus T's of S T is the final temperature of the object. T's of S is the temperature of the surroundings, and we can replace Why not with t not minus t suggests t not is the initial temperature of the object and Tisa best is the temperature of the surroundings. So for this problem, we know the temperature of the surroundings that would be the room temperature is 75 F, and we also know the initial temperature of the object, which is turkey, because it comes out of the oven at 100 85 degrees so we can go ahead and substitute those numbers into our equation. And we have t minus 75 equals 1 85 minus 75 times E raised to the power. Katie, we haven't found k it. Okay, so what? We want to do is use this point that were given The temperature of the turkey is 150 degrees when the time is 30. We're going to use that to help us find the value of K. So we'll substitute 1 50 for the temperature, the final temperature. We can go ahead and subtract the 75 from the 1 85 and that gives us 1 10 and we can put 30 minutes. And for the time, and we'll solve this for K 1 50 minus 75 is 75. Then we're going to divide both sides by 1 10 Oops, that's supposed to say 1/10 and we end up with 75. Divided by 1 10 simplifies to be 15/22 so 15/22 equals each of the 30 K. We take the natural log of both sides and then we divide both sides by 30. So we have K equals natural log of 15/22 divided by 30. So this gives us our model, which for this problem is going to be T minus 75 equals 1 10 times E to the natural log of 15/22 times. 30/30 times T. Okay, so we're going to use that model to help us finish part A, which is to find the temperature when the time is 45 minutes. So here's that model again, and we're going to substitute 45 for the time and this all goes in the calculator very, very carefully, making sure to have all the parentheses in the right place, and we end up with t minus 75 equals approximately 62 degrees. So now we're gonna add 75 to that, and we get approximately 137 degrees, so that tells us that at 45 minutes the turkey would be 137 degrees. Okay, let's move on to part B. So in this part, we're figuring out the time when the temperature is 100 degrees so we can take the same model, which we have right here, and we can substitute 100 for Capital T, and we're gonna sell for lower case T. Okay, so let's subtract. We get 25 and then we're gonna go ahead and divide both sides by 1 10 and 25 divided by 1 10 simplifies to be 5/22 and then we're going to take the natural log on both sides. And now to get t by itself, let's multiply both sides by the reciprocal of this fraction. So we end up with t equals 30 times a natural log of 5/22 divided by the natural log of 15/22. So we also carefully put that into a calculator and we end up with approximately 116 minutes.

View More Answers From This Book

Find Another Textbook

01:16

What is the correct answer?Thanks

Use geometry (not Riemann sums…

01:18

Determine whether the graph of the function provided is a graph of …

04:38

Find a polential function f for the field FFa (dy + 3z)i + (4x …

02:25

Evaluate the integral:r3 + 6r + 2 dc

05:06

hotel chain typically charges $120 per room and rents average of 40 rooms Pe…

01:56

patient's pulse measures 80 bpm 130 bpm_ tnen 90 bpm_ To determine an a…

05:47

Consider the differential equation: 2y" + 18y = 6tan 3x …

01:30

Find all values x = where the function discontinuous For each…

03:18

Find the intervals on which ((x) is increasing, (he intervals on which ((x) …

05:22

Find the first four nonzero terms of the Taylor series for the func…