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Jordan works in a science lab where he is studying the behavior of a certain unstable isotope. He has 240 milligrams of the sample, and the amount of the substance remaining in the sample decreases at a rate of 8% each day. After t days, there are less than 115 milligrams of the substance remaining. Which inequality represents this situation, and after how many days will the amount of the sample be less than 115 milligrams?
A. 115(0.92)t < 240; 10 days
B. 240(0.92)t < 115; 9 days
C. 240(1.08)t < 115; 9 days
D. 115(1.08)t < 240; 8 days

Respuesta :

JeanaShupp

Answer:1) B. 240(0.92)^t < 115; 9 days  is the inequality which represents the situation.

2)After t=9 days the amount of the sample be less than 115 milligrams.

Step-by-step explanation:

The amount of sample which Jordan had in starting a =240 milligrams

rate of decreasing it r =8%=0.08

let t be the number of days the amount of sample took to become 115 milligrams then by the exponential decay  function

[tex]f(t)=a(1-r)^t[/tex] where f(t) =115

[tex]115=240(1-0.08)^t....(1)\\\Rightarrow\frac{115}{240}=(0.92)^t\\\Rightarrow0.479=(0.92)^t\\\text{taking log on both sides ,we get}\\log(0.479)=log((0.92)^t)\\\Rightarrow\ log(0.479)=t\ log(.092)\\\Rightarrow\ -0.735=t(-0.0833)....\text{after solving log values}\\\Rightarrow\ t=\frac{-0.735}{-0.833}\approx8.82 days[/tex]or 9 days(approx ).

therefore, after t=9 days the amount of the sample be less than 115 milligrams.

So ,the right inequality which represent the situation is

B. 240(0.92)^t < 115; 9 days (from (1))

Answer:

B.  240(0.92)t < 115

9 days.

Step-by-step explanation:

We are told that Jordan works in a science lab where he is studying the behavior of a certain unstable isotope. He has 240 milligrams of the sample, and the amount of the substance remaining in the sample decreases at a rate of 8% each day. After t days, there are less than 115 milligrams of the substance remaining. We  are asked to write an inequality to represent this situation.

An exponential function representing decay will be our inequality.Let us write an inequality for this situation.

[tex]240(1-0.8)^{t} <115=240(0.92)^{t} <115[/tex], where 240 is our initial amount of substance, 0.8 is decrease factor.

Now let us solve our inequality to find out number of days.

[tex](0.92)^{t} <\frac{115}{240}[/tex]

Upon taking natural log of both sides of inequality,

[tex]ln(0.92)^{t} <ln\frac{115}{240}[/tex]

[tex]t\ln \left(0.92\right)<ln(\frac{115}{240})[/tex]

[tex]-0.08338160t < -0.73570679[/tex]

[tex]t>8.82[/tex]

Upon rounding up our answer to nearest integer we can see that option B is correct. Therefore, our answer will be option B.

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