Mathematics, 13.07.2019 23:10 lilloser

The number n(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. initially, n(0) = 500, and it is observed that n(1) = 1000. solve for n(t) if it is predicted that the limiting number of people in the community who will see the advertisement is 50,000.

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Answer from: fayth1760

$N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}$

Step-by-step explanation:

The logistic equation is

$N(x)=\frac{c}{1+ae^{-rx}}$

where:

c/(1+a) is the initial value.

c is the limiting value

r is constant determined by growth rate

So we are given that:

N(0)=500 or that c/(1+a)=500

If your not sure about his initial value of c/(1+a) then replace x with 0 in the function N:

$N(0)=\frac{c}{1+ae^{-r \cdot 0}}$

Simplify:

$N(0)=\frac{c}{1+ae^{0}}$

$N(0)=\frac{c}{1+a(1)}$

$N(0)=\frac{c}{1+a}$

Anyways we are given:

$\frac{c}{1+a}=500$ .

Cross multiplying gives you c=500(1+a) .

We are also giving that N(1)=1000 so plug this in:

$N(1)=\frac{c}{1+ae^{-r \cdot 1}}$

Simplify:

$N(1)=\frac{c}{1+ae^{-r}}$

So this means

$1000=\frac{c}{1+ae^{-r}}$

Cross multiplying gives you $c=1000(1+ae^{-r})$

We are giving that c=50000 so we have these two equations to solve:

50000=500(1+a)

and

$50000=1000(1+ae^{-r})$

I'm going to solve 50000=500(1+a) first because there is only one constant variable here,.

50000=500(1+a)

Divide both sides by 500:

100=1+a

Subtract 1 on both sides:

99=a

Now since we have we can find in the second equation:

$50000=1000(1+ae^{-r})$ with a=99

$50000=1000(1+99e^{-r})$

Divide both sides by 1000

$50=1+99e^{-r}$

Subtract 1 on both sides:

$49=99e^{-r}$

Divide both sides by 99:

$\frac{49}{99}=e^{-r}$

Take natural log of both sides:

$\ln(\frac{49}{99})=-r$

Multiply both sides by -1:

$-\ln(\frac{49}{99})=r$

So the function N with all the write values plugged into the constant variables is:

$N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}$

Answer from: Quest

y = 5x + 17

step-by-step explanation:

a perpendicular line will be the opposite reciprocal slope of the given equation. the slope given is -1/5 so the opposite reciprocal is 5

find b:

7 = 5 (-2) + b

7 = -10 + b

17 = b

write equation

y = 5x + 17

Answer from: Quest

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

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