Consider each of the scenarios below. If initial values were provided for each scenario, which could be modeled by an equation of the form y=ax+b? Select all that apply. A. The value of a car depreciates by 10% every year. B. Every 8 hours, half of a drug dosage remains in the body. C. Every week, 3/5 of a radioactive substance remains from the beginning of the week. D. A savings account, which earns no interest, receives a deposit of $500 per month. E. A liter of water evaporates from a swimming pool every day.

[tex]\rm y = L-nx \\y = Final \;volume \; of\; water \;in \;the \; pool \\n= Number \; of \; days\\L= Initial \; amount \; of \; water \\x = Amount \;of \; water\; evaporated \; per \; day[/tex]

So option E is correct

The given scenarios are

A. The value of a car depreciates by 10% every year.

B. Every 8 hours, half of a drug dosage remains in the body.

C. Every week, 3/5 of a radioactive substance remains from the beginning of the week.

D. A savings account, which earns no interest, receives a deposit of $500 per month.

E. A liter of water evaporates from a swimming pool every day.

Equation [tex]\rm y =ax+b[/tex] is the equation of straight line

Out of given options only D and E show the straight line relationship

When am amount P depreciates at a rate of R% per year then the amount [tex]\rm A_n[/tex] after n years can be written as formulated in equation (1)

[tex]\rm A_n = P(1-R)^n.........(1) \\A_n=Amount \; after\; n \; years \\P = Initial \; Amount\\R = Rate\; of \; Depriciation[/tex]

So the equation for depreciation D

[tex]\rm D=P-A_n[/tex]

Clearly this equation is not a linear equation of the form [tex]\rm y =ax+b[/tex]

hence option (A) is ruled out.

Similarly the exponential decay function is given by equation

[tex]\rm A = A_oe^{-\lambda t}\\A = Substance \; remained \;after \;time \; t \\A = Initial\; amount \; of \; substance \\\lambda = 0.693 + T_{1/2}\\T_{1/2} = Half\; life[/tex]

hence we can rule out option (B) and (C)

For option D we can write equation as

[tex]\rm y = 500\times x + I \\y = Final \; amount\\x = Number \; of \; month\\I = Initial \; balance[/tex]

So option D is correct

For option E we can write equation as

[tex]\rm y = L-nx \\y = Final \;volume \; of\; water \;in \;the \; pool \\n= Number \; of \; days\\L= Initial \; amount \; of \; water \\x = Amount \;of \; water\; evaporated \; per \; day[/tex]

So option E is correct

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