Question

1. the table shown gives the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.

time (minutes), x temperature (°f), y
0 179.5
5 168.7
10 152.3
15 141.7
20 135.4
25 123.5
30 116.3
35 113.4
40 106.9
a. use regression on desmos or a graphing calculator to write an exponential function to model the data. round values to 3 decimal places.
b. does this model exponential growth or decay? how can you tell?

1. as a radiology specialist, you use the radioactive substance \iodine - 131\ to diagnose conditions of the thyroid gland. the amount of radioactive substance decays with time. your hospital currently has a 20 - gram supply of iodine - 131. the following function models the number of grams remaining after x days. y = 20(0.917)^x

a. determine the daily percentage decay rate.
b. determine how much substance remains after 5 days. round to the nearest whole number of grams.
c. determine the half - life of iodine - 131 by finding how many days it takes for the 20 - gram supply to decrease to 10 grams. round to the nearest whole number of days. (you can use a graph, table, or guess & check)
d. explain the practical meaning of the horizontal asymptote in this situation.

1. a child currently receives a weekly allowance of $3 and asks their parents for a raise. the parent offers the child two options:

option 1: the child gets a $0.25 raise each week
option 2: the child gets a 5% raise each week
a. write an equation for each option to determine the weekly allowance y after x weeks of raises.
b. use technology to graph your two functions together to compare them. which option is better at first? after how many weeks will the other option begin to offer a bigger allowance?

Explanation:

5.

a.

Step1: Recall decay - rate formula

For an exponential decay function $y = a(1 - r)^x$, where $r$ is the decay - rate. Given $y = 20(0.917)^x$, we set $1 - r=0.917$.

Step2: Solve for $r$

$r = 1 - 0.917=0.083$. Multiply by 100 to get the percentage. So the daily percentage decay rate is $8.3\%$.

Step1: Substitute $x = 5$ into the function

We have the function $y = 20(0.917)^x$. Substitute $x = 5$ into it: $y = 20\times(0.917)^5$.

Step2: Calculate the value

$(0.917)^5=0.917\times0.917\times0.917\times0.917\times0.917\approx0.630$. Then $y = 20\times0.630 = 12.6\approx13$ grams.

Step1: Set up the equation

We want to find $x$ when $y = 10$ in the equation $y = 20(0.917)^x$. So, $10 = 20(0.917)^x$.

Step2: Simplify the equation

Divide both sides by 20: $\frac{10}{20}=(0.917)^x$, which gives $0.5=(0.917)^x$.

Step3: Take the natural - logarithm of both sides

$\ln(0.5)=\ln((0.917)^x)$. Using the property $\ln(a^b)=b\ln(a)$, we get $\ln(0.5)=x\ln(0.917)$.

Step4: Solve for $x$

$x=\frac{\ln(0.5)}{\ln(0.917)}=\frac{- 0.693}{-0.086}\approx8$ days.

Answer:

$8.3\%$

b.