Linear & Exponential Growth
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SAT Math › Linear & Exponential Growth
Solve $2^{x+1}+2^x=12$.
$1$
$3$
$2$
$0$
Explanation
Factor: $2^x(2+1)=12\Rightarrow 3\cdot 2^x=12\Rightarrow 2^x=4\Rightarrow x=2$. The other values do not satisfy the original equation.
Solve $(3^x)^2=27$ for $x$.
$\frac{3}{2}$
$1$
$-\frac{3}{2}$
$3$
Explanation
$3^{2x}=27=3^3$, so $2x=3$ and $x=\frac{3}{2}$. A negative root is extraneous because $3^x>0$ for all real $x$.
A substance decays according to $N(t)=200(0.5)^{t/3}$. What is $N(6)$?
25
50
75
100
Explanation
Here $t/3=6/3=2$, so $N(6)=200(0.5)^2=200\cdot0.25=50$. The other choices result from taking one half-life instead of two or misreading the exponent.
Solve for $x$: $2^{x+1}=16$.
2
3
4
5
Explanation
Since $16=2^4$, set $x+1=4$ to get $x=3$. Other choices come from misreading the exponent or arithmetic mistakes.
A population is modeled by $P(t)=1200(1.08)^t$, where $t$ is measured in years. What is the percent increase per year?
0.08%
0.80%
8%
108%
Explanation
The growth factor is 1.08, which corresponds to an 8% increase. 108% confuses the factor with the rate, and 0.08% or 0.8% are decimal-place errors.
A quantity is modeled by $P(t)=P_0(1+r)^t$. If it triples in 6 time units, what is $r$?
$0.5$
$3^6-1$
$\dfrac{\ln 3}{6}$
$3^{1/6}-1$
Explanation
Tripling means $(1+r)^6=3$, so $r=3^{1/6}-1$. Choice B is the continuous rate, and A and D are linear or gross misinterpretations.
For $g(t)=120(1.2)^t$, by what factor does $g$ increase when $t$ increases by 3?
$1.6$
$2.2$
$3.6$
$1.728$
Explanation
The factor over 3 units is $(1.2)^3=1.728$. Choices B, C, and D incorrectly add percentages or mix factor with rate.
Let $h(x)=40(0.8)^x$. What is $h(3)$?
16
20.48
25.6
32
Explanation
$h(3)=40(0.8)^3=40(0.512)=20.48$. The other options come from using $(0.8)^2$, confusing the base, or guessing.
A function is defined by $g(t) = 120\cdot 2^{t/3}$. By what factor does $g$ increase every 3 units of $t$?
1.33
2
3
200%
Explanation
Increasing $t$ by 3 increases the exponent by 1, multiplying the value by 2. The other choices confuse factor with percent, misread the denominator, or approximate the growth rate rather than the factor.
A population grows according to $P(t) = 500(1.08)^t$, where $t$ is measured in years. What is the percent increase per year?
0.08%
8%
92%
108%
Explanation
The growth factor $1.08$ corresponds to an 8% increase per year. The other choices confuse the decimal rate with a percent, add 100%, or give the complement.