1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Verdich [7]
3 years ago
15

The rate (In mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function 110I

12 +1+ 9 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?
Mathematics
1 answer:
Ksju [112]3 years ago
7 0

Answer:

P is maximum at I = 2

Step-by-step explanation:

Here is the complete question

The rate (in mg carbon/m³/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 100I/(I² + I + 4) where I is the light intensity (measured in thousands of foot candles). For what light intensity P is a maximum?

To find the value of I at which P is maximum, we differentiate P with respect to I and equate it to zero.

So, dP/dI =  d[100I/(I² + I + 4)]/dI

= [(I² + I + 4)d(100I)/dI - 100Id(I² + I + 4)/dI]/(I² + I + 4)²

= [(I² + I + 4)100 - 100I(2I + 1)]/(I² + I + 4)²

= [100I² + 100I + 400 - 200I² - 100I]/(I² + I + 4)²

= [-100I² + 400]/(I² + I + 4)²

=  -100[I² - 4]/(I² + I + 4)²

Since dP/dI = 0,  -100[I² - 4]/(I² + I + 4)² = 0 ⇒ I² - 4 = 0 ⇒ I² = 4 ⇒ I = ±√4

I = ±2

Since I cannot be negative, we ignore the minus sign

To determine if this is a maximum point, we differentiate dP/dI. So,

d(dP/dI)/dI = d²P/dI² = d[-100[I² - 4]/(I² + I + 4)²]/dI

= [(I² + I + 4)²d(-100[I² - 4])/dI - (-100[I² - 4])d(I² + I + 4)²/dt]/[(I² + I + 4)²]²

= [(I² + I + 4)²(-200I) + 100[I² - 4]) × (2I + 1) × 2(I² + I + 4)]/(I² + I + 4)⁴

= [-200I(I² + I + 4)² + 200[I² - 4])(2I + 1)(I² + I + 4)]/(I² + I + 4)⁴

= [-200(I² + I + 4)[I(I² + I + 4) - [I² - 4])(2I + 1)]]/(I² + I + 4)⁴

= [-200(I² + I + 4)[I³ + I² + 4I - I² + 4])(2I + 1)]]/(I² + I + 4)⁴

= [-200(I² + I + 4)[I³ + 4I + 8])(2I + 1)]]/(I² + I + 4)⁴

Substituting I = 2 into d²P/dI², we have

= [-200(2² + 2 + 4)[2³ + 4(2) + 8])(2(2) + 1)]]/(2² + 2 + 4)⁴

= [-200(4 + 2 + 4)[8 + 8 + 8])(4 + 1)]]/(4 + 2 + 4)⁴

= [-200(10)[24](5)]]/(10)⁴

= -240000/10⁴

= -24

Since d²P/dI² = -24 < 0 at I = 2,  this shows that it I = 2 is a maximum point.

So, P is maximum at I = 2

You might be interested in
The perimeter of a cross section of a piece of lumber is 16 1/2in. The length is twice the width. Find the actual dimensions of
Lorico [155]
<span>the perimeter of the cross section is 16.5 inches.
the perimeter of the cross section is 2*L plus 2*W where L is the length of the cross section and W is the width of the cross section.
since L = 2*W, the perimeter becomes 4*W plus 2*W = 6*W.

16.5 / 6 = 2.75 inches.
W = 2.75 inches.
L = 5.5 inches.
P = 2*L + 2*W = 2*5.5 + 2*2.75 = 11 + 5.5 = 16.5 </span>
6 0
3 years ago
Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must 7 f(x) dx 4 lie?Which property
mr Goodwill [35]

Answer:

3m ≤ ∫ f(x) dx ≤ 3M, at limit of b, a

Step-by-step explanation:

Like the question asked, which property of integral was used.

Property 8 of integrals was the basis upon which the question was solved.

6 0
3 years ago
A farmer uses 36 feet of fencing to build a rectangular rabbit pen. The function A(w)=18w−w2 can be used to find the area of the
Masja [62]

Answer:

Step-by-step explanation: D

3 0
3 years ago
Help me pleaseee!!<br> Would appreciate ^^<br> **BRAINLIST, thanks and points***
olga2289 [7]

Answer:

4900

Step-by-step explanation:

because it is square root of 70 and it means

sol?

square of 70 =70²=70×70=4900

5 0
3 years ago
Read 2 more answers
Evaluate/find all “real” solutions to m squared +4 =0
laiz [17]

Answer:

There is no real solution

Step-by-step explanation:

m^2 = -4

to find m would be a complex number and that would be

m = 2i

but you don't need to find that

6 0
3 years ago
Other questions:
  • Solve for x.<br> z=(x-7)k
    15·1 answer
  • A pilot in an apache rescue helicopter, at an altitude of 1200 feet, spots his two soldiers on the ground. the angles of depress
    9·1 answer
  • PLEASE HELP Consider these dimensions for a cylindrical aquarium.
    11·2 answers
  • Which is greater 7/8 or 5/10
    14·2 answers
  • What is a function exponential
    15·2 answers
  • Y2 + 8zx - 2xy -4yz factorise ?
    6·1 answer
  • What is the answer NEED HELP ASAP<br><br> b-14=9 WHO ever answers gets 10 points
    12·2 answers
  • 8a^3-36a^2+54a-27-64p^3​
    15·1 answer
  • I have 10 apples and lost 5 then got 5 more apples from my mom. My mom also gave me 20 oranges so i ate 12 oranges and 5 apples
    8·2 answers
  • Find the value of n:<br> <img src="https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7Bn%7D%20%3D%5Cfrac%7B15%7D%7B6%7D" id="TexFormula1"
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!