For what value or values of x = 7

One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number o

Zielflug [23.3K]

To be clear, the given relation between time and female population is an integral:
 $t = \int { \frac{P+S}{P[(r - 1)P - S]} } \,
dP$

The problem says that r = 1.2 and S = 400, therefore substituting:
 $t = \int { \frac{P+400}{P[(1.2 - 1)P - 400]}
} \, dP$ 

= $
\int { \frac{P+400}{P(0.2P - 400)} } \, dP$

In order to evaluate this integral, we need to write this rational function in a simpler way:
 $\frac{P+400}{P(0.2P - 400)} = \frac{A}{P} +
\frac{B}{(0.2P - 400)}$ 

where we need to evaluate A and B. In order to do so, let's calculate the LCD:
 $\frac{P+400}{P(0.2P - 400)} = \frac{A(0.2P -
400)}{P(0.2P - 400)} + \frac{BP}{P(0.2P - 400)}$ 

the denominators cancel out and we get:
P + 400 = 0.2AP - 400A + BP
 = P(0.2A + B) - 400A

The two sides must be equal to each other, bringing the system:
 $\left \{ {{0.2A + B = 1} \atop {-400A =
400}} \right.$ 

Which can be easily solved:
 $\left \{ {{B=1.2} \atop {A=-1}} \right.$ 

Therefore, our integral can be written as:
 $t = \int { \frac{P+400}{P(0.2P - 400)} } \,
dP = \int {( \frac{-1}{P} + \frac{1.2}{0.2P-400} )} \, dP$ 
= $- \int { \frac{1}{P} \, dP +
1.2\int { \frac{1}{0.2P-400} } \, dP$ 
= $- \int { \frac{1}{P} \, dP +
6\int { \frac{0.2}{0.2P-400} } \, dP$ 
= - ln |P| + 6 ln |0.2P - 400| + C

Now, let’s evaluate C by considering that at t = 0 P = 10000:
0 = - ln |10000| + 6 ln |0.2(10000) - 400| + C
C = ln |10000| - 6 ln |1600|
C = ln (10⁴) - 6 ln (2⁶·5²)
C = 4 ln (10) - 36 ln (2) - 12 ln (5) 
 Therefore, the equation relating female population with time requested is:
t = - ln |P| + 6 ln |0.2P - 400| + 4 ln (10) - 36 ln (2) - 12 ln (5)

8 0

3 years ago

I just need help on the question circled below. (I need the answer ASAP.)

77julia77 [94]

2.5+ 2.5= 5 miles i think

6 0

3 years ago

A rectangular playground is 10m longer than it is wide.The area of the playground is 1400m² Calculate the width and length of th

Bezzdna [24]

Answer:

$1400 = x(x+10)= x^2 +10x$

We can rewrite this expression like this:

$x^2 +10 x -1400 =0$

And we can use the quadratic formula given by:

$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

Where $a =1 , b=10 , c= -1400$

And replacing we got:

$x = \frac{-10 \pm \sqrt{10^2 -4(10)(-1400)}}{2*1}$

And solving we got:

$x_1 =32.75 , x_2 = -42.75$

And since the value can't be negative the answer would be x = 32.75 and the value of y = 32.75+10 =42.75

Step-by-step explanation:

For this case we know that we have a rectangular playground and the area can be founded with this formula:

$A = xy$

Where x represent the width and y the length. From the problem we know that A =1400 m^2 and the heigth is 10m longer than the wide so we can write this condition as:

$y = 10 +x$

And replacing this formula into the area we got:

$1400 = x(x+10)= x^2 +10x$

We can rewrite this expression like this:

$x^2 +10 x -1400 =0$

And we can use the quadratic formula given by:

$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

Where $a =1 , b=10 , c= -1400$

And replacing we got:

$x = \frac{-10 \pm \sqrt{10^2 -4(10)(-1400)}}{2*1}$

And solving we got:

$x_1 =32.75 , x_2 = -42.75$

And since the value can't be negative the answer would be x = 32.75 and the value of y = 32.75+10 =42.75

8 0

3 years ago

Read 2 more answers

Simplify the expression: x2+x(x-5) -x+7

Veronika [31]

Hi mate!

The answer after simplifying is: 2x^2-6x+7

Please let me know if you need further assistance! Have a terrific evening.
~Brooke❤️

8 0

3 years ago

Write a linear equation for the line with slope = - 1/4 going through the point: (0, 0)

Serjik [45]

The equation of a line is written as y =Mx + b, where m is the slope and b is the y intercept, which is the y value when x = 0.

You are told the slope is -1/4, so replace m with -1/4. You are also given the point (0,0) which means when x is 0, y is also 0, so replace b with 0.

The equation becomes y= -1/4x + 0

Because y is 0 you aren’t adding anything to the equation so you can remove it from the equation.

Answer is y = -1/4x

5 0

3 years ago

Read 2 more answers

For what value or values of x = 7 < 3x + 2?