23. In biology, the function defined by g(x)=5x/2+x
gives the growth rate of a population for x units of available food. (Source: Smith, J. Maynard,
Models in Ecology, Cambridge University Press.)
(a) What amount of food (in appropriate units) would produce a growth rate of 3 units of
growth per unit of food?
(b) What is the growth rate if no food is available?
A. 3units
B. 0units
(a) To find the amount of food that would produce a growth rate of 3 units of growth per unit of food, we need to solve for x in the equation g(x) = 3.
g(x) = 5x/2 + x
Set g(x) equal to 3:
3 = 5x/2 + x
To simplify the equation, we can multiply every term by 2:
6 = 5x + 2x
Combine like terms:
6 = 7x
Now, divide both sides of the equation by 7 to solve for x:
6/7 = x
Therefore, the amount of food that would produce a growth rate of 3 units of growth per unit of food is 6/7 units.
(b) If no food is available (x = 0), we can substitute this value into the equation to find the growth rate:
g(x) = 5x/2 + x
g(0) = 5(0)/2 + 0
= 0/2 + 0
= 0 + 0
= 0
Therefore, the growth rate when no food is available is 0 units of growth per unit of food.
To find the answer to both parts of the question, we need to plug in the given values into the function g(x)=5x/2+x. Let's solve each part step by step:
(a) What amount of food would produce a growth rate of 3 units of growth per unit of food?
To find the amount of food that produces a growth rate of 3 units of growth per unit of food, we need to solve the equation g(x) = 3. Let's substitute and solve:
3 = 5x/2 + x
First, let's multiply both sides of the equation by 2 to eliminate the fraction:
6 = 5x + 2x
Combine like terms:
6 = 7x
Divide both sides of the equation by 7 to solve for x:
x = 6/7
Therefore, the amount of food that produces a growth rate of 3 units of growth per unit of food is 6/7 units of food.
(b) What is the growth rate if no food is available?
To find the growth rate when no food is available, we need to substitute x = 0 into the equation g(x) = 5x/2 + x:
g(0) = 0/2 + 0
g(0) = 0
Therefore, the growth rate when no food is available is 0 units of growth per unit of food.