Answer:
is -12.9<q
Step-by-step explanation:
<h3>Given</h3>
p'(t) = kp²
p(0) = 12; p(10) = 24
<h3>Find</h3>
a) p(t)
b) t such that p(t) = 48
c) the behavior of p(t) after the time of part b
<h3>Solution</h3>
a) The differential equation is separable, so can be solved by separating the variables and integrating.
![\displaystyle\frac{d}{dt}p(t)=k\cdot p(t)^{2}\\\\\int{p^{-2}}\,dp=\int{k}\,dt\\\\-p^{-1}=kt+C\\\\p=\frac{-1}{kt+C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%7D%7Bdt%7Dp%28t%29%3Dk%5Ccdot%20p%28t%29%5E%7B2%7D%5C%5C%5C%5C%5Cint%7Bp%5E%7B-2%7D%7D%5C%2Cdp%3D%5Cint%7Bk%7D%5C%2Cdt%5C%5C%5C%5C-p%5E%7B-1%7D%3Dkt%2BC%5C%5C%5C%5Cp%3D%5Cfrac%7B-1%7D%7Bkt%2BC%7D)
Plugging in the given boundary conditions, we can solve for k and C to find
![p(t)=\dfrac{240}{20-t}](https://tex.z-dn.net/?f=p%28t%29%3D%5Cdfrac%7B240%7D%7B20-t%7D)
b) The population doubles when the time to t=20 is cut in half. The first doubling occurred in 10 years; the second one will occur in half that time, 5 years. There will be 48 alligators in the swamp in 2003.
c) The population doubles again in half the time of the previous doubling, so is predicted to be infinite in 2008.
![x = 47](https://tex.z-dn.net/?f=x%20%3D%2047)
Step-by-step explanation:
The 3rd interior angle of the triangle can be written as
![\angle{3} = 180 - 45 - x = 135 - x](https://tex.z-dn.net/?f=%5Cangle%7B3%7D%20%3D%20180%20-%2045%20-%20x%20%3D%20135%20-%20x)
The angle above is supplementary to the angle (2x - 2), therefore
![(135 - x) + (2x - 2) = 180 \Rightarrow 133 + x = 180](https://tex.z-dn.net/?f=%28135%20-%20x%29%20%2B%20%282x%20-%202%29%20%3D%20180%20%5CRightarrow%20133%20%2B%20x%20%3D%20180)
Solving for x, we get
![x = 47](https://tex.z-dn.net/?f=x%20%3D%2047)
Answer:
$192
Step-by-step explanation:
The cost function is given as:
C(x)=18x+240
The price function is given as:
p(x)= 90 - 3x
The revenue R(x) is the product of the price and the number of products. It is given by:
R(x) = xp(x) = x(90 - 3x) = 90x - 3x²
The profit P(x) is the difference between the revenue and the cost of production. Therefore:
P(x) = R(x) - C(x) = 90x - 3x² - (18x + 240) = 90x - 3x² - 18x - 240
P(x) = -3x² + 72x - 240
The standard equation of a quadratic equation is ax² + bx + c. The function has a maximum value at x = -b/2a
Since P(x) = -3x² + 72x - 240, the maximum profit is at:
x = -72/2(-3) = 12
at x = 12, the profit is:
P(12) = -3(12)² + 72(12) - 240 = -432 + 864 - 240 = $192
Two consecutive integers are -4-9 and 1(-13)