Questions: I. (5 points) The growth rate for a fish depends on its weight. For a certain fish population, the growth rate G as a function of its weight w is G(w)=-w^2+4w The weight, w is measured in ounces and G is measured in ounces per year. a. Is the function G(w) concave up or concave down, and why? i. Answer: ii. Reason: b. At what weight is the fish growing the fastest (maximized), and what is the maximum growth rate? Show all your steps and include all units. i. Weight = ii. Growth rate = c. Assuming that the end of life occurs when growth stops (G(w)=0), at what weight will death occur? Show all your steps and include all units. i. Value only: ii. Units only:

I. (5 points) The growth rate for a fish depends on its weight. For a certain fish population, the growth rate G as a function of its weight w is
G(w)=-w^2+4w

The weight, w is measured in ounces and G is measured in ounces per year.
a. Is the function G(w) concave up or concave down, and why?
i. Answer:
ii. Reason:
b. At what weight is the fish growing the fastest (maximized), and what is the maximum growth rate? Show all your steps and include all units.
i. Weight =
ii. Growth rate =
c. Assuming that the end of life occurs when growth stops (G(w)=0), at what weight will death occur? Show all your steps and include all units.
i. Value only:
ii. Units only:

Transcript text: I. (5 points) The growth rate for a fish depends on its weight. For a certain fish population, the growth rate $G$ as a function of its weight $w$ is \[ G(w)=-w^{2}+4 w \] The weight, $w$ is measured in ounces and $G$ is measured in ounces per year. a. Is the function $G(w)$ concave up or concave down, and why? i. Answer: ii. Reason: b. At what weight is the fish growing the fastest (maximized), and what is the maximum growth rate? Show all your steps and include all units. i. Weight $=$ ii. Growth rate $=$ c. Assuming that the end of life occurs when growth stops $(G(w)=0)$, at what weight will death occur? Show all your steps and include all units. i. Value only: ii. Units only: Page 2 of 5 Initials:

Solution

Solution Steps

Solution Approach

a. To determine if the function $ G(w) = -w^2 + 4w $ is concave up or concave down, we need to look at the second derivative of the function. If the second derivative is positive, the function is concave up; if it is negative, the function is concave down.

b. To find the weight at which the fish is growing the fastest, we need to find the maximum of the function $ G(w) $. This involves finding the first derivative, setting it to zero to find critical points, and then using the second derivative test to confirm if it is a maximum. The maximum growth rate is the value of $ G(w) $ at this weight.

c. To find the weight at which growth stops, we need to solve the equation $ G(w) = 0 $ for $ w $.

Step 1: Determine Concavity

To determine the concavity of the function $ G(w) = -w^2 + 4w $, we calculate the second derivative: \[ G''(w) = -2 \] Since $ G''(w) < 0 $, the function is concave down.

Step 2: Find Maximum Growth Rate

Next, we find the weight at which the fish is growing the fastest by setting the first derivative to zero: \[ G'(w) = 4 - 2w = 0 \implies w = 2 \] To find the maximum growth rate, we evaluate $ G(w) $ at $ w = 2 $: \[ G(2) = -2^2 + 4 \cdot 2 = 4 \] Thus, the maximum growth rate occurs at a weight of $ w = 2 $ ounces, with a growth rate of $ G = 4 $ ounces per year.

Step 3: Find Weight at Which Growth Stops

To find the weight at which growth stops, we solve the equation $ G(w) = 0 $: \[ -w^2 + 4w = 0 \implies w(w - 4) = 0 \] This gives us the solutions $ w = 0 $ and $ w = 4 $. Therefore, growth stops at weights of $ 0 $ ounces and $ 4 $ ounces.

Final Answer

a. The function is concave down.
b. Weight $ = 2 $ ounces; Growth rate $ = 4 $ ounces per year.
c. Weight at which growth stops: $ 0 $ ounces and $ 4 $ ounces.

\[ \boxed{ \begin{align_} \text{a.} & \text{ Concave down} \\ \text{b.} & \text{ Weight } = 2 \text{ ounces}, \text{ Growth rate } = 4 \text{ ounces per year} \\ \text{c.} & \text{ Weight } = 0 \text{ ounces and } 4 \text{ ounces} \end{align_} } \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!