Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

The temperature T, in degrees Fahrenheit, of a yam after sitting in a hot oven for t minutes is given by the function T(t)=350-280e-0.2t

(a) What is the initial temperature of the yam, before it is put in the oven?

(b) Given that over time the temperature of the yam will approach the temperature inside the oven, use a limit to determine the temperature of the oven.

(c) How long will it take for the yam to be within 5 degrees Fahrenheit of the temperature of the oven?

(d) The first derivative of T(t) measures the rate of change of the temperature of the yam. The second derivative of T(t) measures the rate of change of the rate of change of the temperature of the yam. Use T'(t) and T''(t) to argue that the temperature of the yam increases at a decreasing rate. This statement is related to the odd saying “Cold water boils faster.” How?

Short Answer

Expert verified

Part(a)T(t)=70°Part(b)T(t)=350°Part(c)20minutesPart(d)thetemperatureoftheyamincreasesatadecreasingrate.

Step by step solution

01

Part(a) Step 1. Given Information

The given function isT(t)=350-280e-0.2t

02

Part(a) Step 2. Calculation

Substitute t as 0 in the given function to find the initial temperature.

T(t)=350-280e-0.2tT(0)=350-280e-0.2(0)=350-280=70°

03

Part(b) Step 1. Calculation

Substitute t=in the given function.

T(t)=350-280e-0.2tT()=350-280e-0.2()=350-0=350°

04

Part(c) Step 1. Calculation

Here, we have,

T(t)=350-280e-0.2t350-5=350-280e-0.2t345=350-280e-0.2t280e-0.2t=5e-0.2t=5280-0.2tln(e)=ln5280t=ln5280-0.2t=20minutes

05

Part(d) Step 1. Calculation

The model for T'(t) is as follows,

T'(t)=ddt350-280e-0.2t=-280(-0.2)e-0.2t=56e-0.2t

And the model for T''(t) is as follows,

T''(t)=56ddte-0.2t=56(-0.2)e-0.2t=-11.2e-0.2t

As the coefficient is decreasing naturally, therefore we can conclude that the temperature of the Yam increases at a decreasing rate.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free