University:
Morehouse College
Course:
HMTH 105 | College Algebra 1
Academic year:

2024
Views:

315

Pages:

4
Author:

Reynaldo Wong

Question 1 What is the difference between linear growth and exponential growth? Answer 1 Linear growth is always at the same rate, and exponential growth increases in speed over time. Answer 2 L. Linear growth occurs by adding the same numbers, and exponential growth occurs by multiplying the same numbers. 2. Linear growth is a slower form of growth, and exponential is faster form. 3. Linear equations dont Answer 3 If growth is plotted in a diagram and it resembles a straight line, this is called linear growth. Additive processes produce linear growth. Exponential growth is much faster than linear growth. Exponential growth occurs in cases of unrestrained positive feedback. Change is multiplied during each time period. Question 2 Find the exponential model y ae that fits the points (0, 2) and (4,3) Answer 1 Use the first point to fina a. y = ache 2 = a * e ^ (ba) 2 = a * 1 a=2 Use the second point to find b y = 2e ^ (3x) 3 = 2e ^ (4t) sigma/9 = e ^ 40 a * 3/2 = ln(e ^ (4t)) 4b = ln(3/2) b = 1/4 * ln(3/2) equation for the exponential model. y = 2e ^ (d/4 * ln(1/4)) -2 tau = 0.012936 Therefore the population t years after 1990 is, P(t) = 29.7 b) The double of 29.76 million is 59.52 million. So put P(t) 59.52 in P(t) = 29.76e ^ (0.0125t) we get 59.52 = 29.76c ^ 6 Rightarrow e ^ (0.01203t) = 2 0.0129364 In (2) ⇒t= In (2) 0.012936 Rightarrow t = 53.5828 Hence the population will double in 53.5828 years. c) Since 2010-1990-20, therefore to predict the population in 2010, put t = 20 in P(t) = 29.76e ^ (0.01293t) we get P(20) = 29.76e ^ 0 (20) 38.5472 million The population in California in 2010 is 37.3 million, therefore the actual population is approximately 1 million less than the obtained population Question 5 Newton's law of cooling indicates that the temperature of a warn object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T'(t)Ta (ToTae in this model, T, represents the temperature of the surrounding air. To represents the initial temperature of the object and t is the time after the object starts cooling. The value of k is the cooling rate and is a constant related to the physical properties the object. A cake comes out of the oven at 335F and is placed on a cooling rack in a 70F kitchen. After checking the temperature several minutes later, it is determined that the cooling rate k is 0.050. Write a function that models the temperature T(t) of the cake t minutes after being removed from the oven. Answer 1 Given The initial temperature (TO) of the cake is 335°F. The tempearture (Ta) of the cake when place in cooling tray is 70°F. The cooling rate (k) is 0.050. Known fact By the Newton's law of cooling, the temperature is modeled as T(t)T(Ta-Tale where, Ta is the temperature surronded by air. To is the initial temperature t is the time after the object storts cooling. k is the cooling rate. Calculation: The function that models the temperature of the cake t minutes after being removed from the oven is computed as follows. Substitute To 335F, Ta 70F and k=0.050 T(t)Ta+(To+Tale T(1)70+(335-70)( 70+265 (0.000) Answer: The function that models the temperature of the cake t minutes after it is removed from the oven is 7'(t)=70+265 (0.000)