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3 years ago

HELP ILL GIVE BRAINLIEST

Mathematics

1 answer:

statuscvo [17]3 years ago

5 0

A) 4.5×10¹¹ B) 3.5×1000

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In 2012, the world population was 7.02 billion. The birth rate had stabilized to 134 million per year and is projected to remain

xeze [42]

Answer:

$(a)\ \frac{dP}{dt} = 0.078- 0.000857t$

$(b)\ P = 0.078t- 0.0004285t^2 + 7.02$

$(c)\ 9.37\ billion$

Step-by-step explanation:

Given

In 2012

$P(t) = 7.02\ billion$

$R_1 = 134\ million/year$ --- Birth rate

$R_2 = 56\ to\ 80$ -- Death rate from '12 to '40

Solving (a): Equation for population from 2012

In 2012, we have the addition in population to be:

$\triangle P= Birth - Death$

$\triangle P= 134 - 56$

$\triangle P = 78$ million

From 2012 to 2040, we have the death rate per year to be:

$Rate/yr = \frac{80 - 56}{2040 - 2012}$

$Rate/yr = \frac{24}{28}$

$Rate/yr = \frac{6}{7}$ million/year

So, the differential equation is:

$\frac{dP}{dt} = \triangle P - Rate/yr * t$ where t represents time

$\frac{dP}{dt} = 78 - \frac{6}{7} * t$

$\frac{dP}{dt} = 78 - \frac{6t}{7}$

Express in millions

$\frac{dP}{dt} = \frac{78}{1000} - \frac{6t}{7000}$

$\frac{dP}{dt} = 0.078- 0.000857t$

Solving (b): The solution to the equation in (a)

$\frac{dP}{dt} = 0.078- 0.000857t$

Make dP the subject

$dP = (0.078- 0.000857t)dt$

Integrate both sides

$\int dP = \int (0.078- 0.000857t)dt$

$P = \int (0.078- 0.000857t)dt$

This gives:

$P = 0.078t- \frac{0.000857t^2}{2} + c$

$P = 0.078t- 0.0004285t^2 + c$

Solve for c.

When t = 0; P = 7.02

So, we have:

$7.02 = 0.078*0- 0.0004285*0^2 + c$

$7.02 =0-0 + c$

$7.02 =c$

$c = 7.02$

So:

$P = 0.078t- 0.0004285t^2 + c$

$P = 0.078t- 0.0004285t^2 + 7.02$

Solving (c): The population in 2050

We have:

$P = 0.078t- 0.0004285t^2 + 7.02$

In 2050, the value of t is:

$t = 2050 - 2012$

$t = 38$

So, the expression becomes

$P = 0.078*38- 0.0004285*38^2 + 7.02$

$P \approx 9.37\ billion$

3 0

3 years ago

A 40-foot tree casts a 12-foot shadow. How long is the shadow of a 5-foot-tall person?

Reptile [31]

Answer:

Step-by-step explanation:

For this type of problem, we need to think of it as a ratio.

The first tree to shadow ratio would be 40:12 this can be simplified to 20:6 which can be simplified to 10:3.

The second ratio is 5:x. I say x because it's an unknown number. Since 5 is half of ten, then the x would be half of three. Therefore the answer is D.

6 0

3 years ago

Read 2 more answers

Prove that tan2A is equal to 2tanA/1-tan^2A

Neko [114]

$\tan(2\alpha)=\dfrac{2\tan \alpha }{1-\tan^2 \alpha}\\\\\\\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{\dfrac{2\sin\alpha}{\cos\alpha}}{1-\dfrac{\sin ^2\alpha}{\cos^2\alpha}}\\\\\\\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{\dfrac{2\sin\alpha}{\cos\alpha} }{\dfrac{\cos^2\alpha}{\cos^2\alpha}-\dfrac{\sin ^2\alpha}{\cos^2\alpha}}\\\\\\\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{\dfrac{2\sin\alpha}{\cos\alpha} }{\dfrac{\cos^2\alpha-\sin^2 \alpha}{\cos^2\alpha}}$

$\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{2\sin\alpha}{\cos\alpha} \cdot\dfrac{\cos^2\alpha}{\cos^2\alpha-\sin^2 \alpha}\\\\\\\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{2\sin\alpha\cos\alpha}{\cos^2\alpha-\sin^2 \alpha} \\\\\\\dfrac{\sin(2\alpha)}{\cos(2\alpha)}=\dfrac{\sin(2\alpha)}{\cos(2\alpha)}$