Can someone simplify this?? ty! -(x-8)²+4

Mathematics

1 answer:

Sindrei [870]2 years ago

7 0

Answer:

-x² + 16x - 60

Step-by-step explanation:

Given expression:

-(x - 8)² + 4

Step 1. Insert brackets to isolate the "-" from the (x - 8)².

⇒ -(x - 8)² + 4

⇒ -[(x - 8)²] + 4

Step 2. Use the formula (a - b)² = a² - 2ab + b²

⇒ -[(x - 8)²] + 4

⇒ -[x² - 2(x)(8) + 8²] + 4

Step 3. Open the parenthesis

⇒ -[x² - 2(x)(8) + 8²] + 4

⇒ -x² + 2(x)(8) - 8² + 4

Step 4. Simplify as needed

⇒ -x² + 16(x) - 64 + 4

⇒ -x² + 16x - 60

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Two years ago the population of a town was 40000. The population of the town at present has reached 44100. Calculate the populat

LenaWriter [7]

Answer:

5% annual population growth rate

Step-by-step explanation:

Let the percent the population grows by be $X\%$ . The total population, $f(x)$ , after $t$ years can be modeled by the function:

$f(x)=40,000\cdot (\frac{X}{100}+1)^t$

Why?

Let's take a look at a simple example. If we said a number $n$ grew by 10%, we could represent the number after it grew by multiplying $n$ by $1.10$ . This is because growing by 10% is equivalent to taking $100\%+10\%=110\%$ of that number and we convert a percentage to a decimal by dividing by 100.

Therefore, if the population grew $X\%$ , we would divide it by 100 to convert it to a decimal, then add 1 (100%) and raise to the power of $t$ (number of years) to multiply by the initial population of 40,000 to get the total population after $t$ years.

Since the population of the town after two years is 44,100, substitute $f(x)=44,100$ and $t=2$ into $f(x)=40,000\cdot (\frac{X}{100}+1)^t$ :

$44,100=40,000\cdot (\frac{X}{100}+1)^2,\\\\(\frac{X}{100}+1)^2=\frac{44,100}{40,000},\\\\(\frac{X}{100}+1)^2=1.1025,\\\\(\frac{X}{100}+1)^2=\pm \sqrt{1.1025},\\\\\begin{cases}\frac{X}{100}+1=1.05,\frac{X}{100}=0.05, X=\boxed{5\%},\\*\text{negative case is extraneous since X must be positive}\end{cases}$