Question

10
is the data best represented by a linear or exponential model? write the equation used to model the data. (a.fgr.9.5)

year

estimated population

----

----

7

70

8

67

9

64

10

61

19

34

a) exponential: $y = 70(0.96)^x$

b) linear: $y = -3x + 91$
c) exponential: $y = -3(0.96)^x$
d) linear: $y = -3x + 70$

Explanation:

Step1: Check linear differences

Calculate differences between consecutive $y$-values:
$70-67=3$, $67-64=3$, $64-61=3$, $61-34=27$
Wait, correct consecutive pairs: Year7→8: $70-67=3$; Year8→9: $67-64=3$; Year9→10: $64-61=3$; Year10→19: $61-34=27$. But for the first 4 points, the common difference is $-3$ (since $y$ decreases by 3 when $x$ increases by 1). Now test linear equation $y=-3x+91$:
For $x=7$: $y=-3(7)+91=-21+91=70$ ✔️
For $x=8$: $y=-3(8)+91=-24+91=67$ ✔️
For $x=9$: $y=-3(9)+91=-27+91=64$ ✔️
For $x=10$: $y=-3(10)+91=-30+91=61$ ✔️
For $x=19$: $y=-3(19)+91=-57+91=34$ ✔️

Step2: Reject exponential models

Test option A: $y=70(0.96)^x$
For $x=7$: $y=70(0.96)^7\approx70\times0.751\approx52.57
eq70$ ❌
Test option C: $y=-3(0.96)^x$
For $x=7$: $y=-3(0.96)^7\approx-3\times0.751\approx-2.25
eq70$ ❌

Step3: Verify remaining linear model

Test option D: $y=-3x+70$
For $x=7$: $y=-3(7)+70=-21+70=49
eq70$ ❌
Only option B's equation fits all data points.

Answer:

B) Linear: $y = -3x + 91$