Question

the equation of an ellipse is shown below. what are the foci of this ellipse?
\\(\frac{(x - 12)^2}{289}+\frac{(y - 3)^2}{64}=1\\)
(27,3) and (-3,3)
(18,12) and (-12,12)
(3,27) and (3,-3)
(12,18) and (12,-12)
question 22 = multiple choice question
if the population increases by 2.5% per year, what percentage does it change in 5 years?
13.1% 13.1%
7.5% 7.5%
16.3% 16.3%
12.5% 12.5%

Explanation:

Step1: Identify ellipse parameters

The standard - form of an ellipse is $\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1$. Here $h = 12,k = 3,a^{2}=289\Rightarrow a = 17,b^{2}=64\Rightarrow b = 8$.

Step2: Calculate $c$

Use the formula $c=\sqrt{a^{2}-b^{2}}=\sqrt{289 - 64}=\sqrt{225}=15$.

Step3: Find foci

The foci of the ellipse $\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1$ are $(h\pm c,k)$. So the foci are $(12 + 15,3)$ and $(12-15,3)$ i.e., $(27,3)$ and $(-3,3)$.

Step1: Use compound - growth formula

Let the initial population be $P$. After $n = 5$ years with a growth rate $r=0.025$ per year, the population $A=P(1 + r)^n=P(1 + 0.025)^5$.

Step2: Calculate the factor of growth

$(1 + 0.025)^5=1.025^5\approx1.131$.

Step3: Find percentage change

The percentage change is $(1.131 - 1)\times100\%=13.1\%$.

Answer:

$(27,3)$ and $(-3,3)$