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3 years ago

If PQ=8 and Q lies at -13 where could P be located?

Mathematics

1 answer:

choli [55]3 years ago

7 0

Answer:

In the given figure the point on segment PQ is twice as from P as from Q is. What is the point? Ans is (2,1).

Step-by-step explanation:

There is really no need to use any quadratics or roots.

( Consider the same problem on the plain number line first. )

How do you find the number between 2 and 5 which is twice as far from 2 as from 5?

You take their difference, which is 3. Now splitting this distance by ratio 2:1 means the first distance is two thirds, the second is one third, so we get

4=2+23(5−2)

It works completely the same with geometric points (using vector operations), just linear interpolation: Call the result R, then

R=P+23(Q−P)

so in your case we get

R=(0,−1)+23(3,3)=(2,1)

Why does this work for 2D-distances as well, even if there seem to be roots involved? Because vector length behaves linearly after all! (meaning |t⋅a⃗ |=t|a⃗ | for any positive scalar t)

Edit: We'll try to divide a distance s into parts a and b such that a is twice as long as b. So it's a=2b and we get

s=a+b=2b+b=3b

⇔b=13s⇒a=23s

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The world's population was 5.51 billion on January 1, 1993 and 5.88 billion on January 1, 1998. Assume that at any time the popu

larisa86 [58]

Answer:

The world's population should reach 8 billion during the year 2021.

Step-by-step explanation:

The world population can be modeled by the following equation:

$\frac{dP}{dt} = r$

It's solution is:

$P(t) = P(0)e^{rt}$

In which P(t) is the population in t years ater 1993, in billions of people, P(0) is the initial population(in 1993) and r is the growth rate.

5.51 billion on January 1, 1993

This means that P(0) = 5.51.

5.88 billion on January 1, 1998.

1998 - 1993 = 5

This means that P(5) = 5.88

We use this as a mean to find the value of r.

$P(t) = P(0)e^{rt}$

$5.88 = 5.51e^{5r}$

$e^{5r} = \frac{5.88}{5.51}$

$e^{5r} = 1.06715$

$\ln{e^{5r}} = \ln{1.06715}$

$5r = \ln{1.06715}$

$r = \frac{\ln{1.06715}}{5}$

$r = 0.013$

Assume that at any time the population grows at a rate proportional to the population at that time. In what year should the world's population reach 8 billion

t yers after 1993, in which t is found when P(t) = 8. So

$P(t) = 5.51e^{0.013t}$