Let’s delve into solving these questions.

Question 5: Writing the Equation for Total Area $A(w)$

The rectangular garden is divided into 2 smaller gardens of equal size, and the fencing is 600 feet total. The fencing includes:

1. Two vertical sections for the width ($w$).
2. Three horizontal sections for the length ($l$).

The total fencing equation is: $2w + 3l = 600$

From this, solve for $l$ in terms of $w$: $l = \frac{600 - 2w}{3}$

The total area $A(w)$ of the garden is: $A(w) = w \cdot l$

Substituting $l = \frac{600 - 2w}{3}$ into the area equation: $A(w) = w \cdot \frac{600 - 2w}{3}$

Simplify: $A(w) = \frac{600w - 2w^2}{3}$

Thus, the equation for the total area is: $A(w) = \frac{600w - 2w^2}{3}$

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Question 6: Maximizing the Area

To find the dimensions for the maximum area, maximize $A(w)$. The formula is: $A(w) = \frac{600w - 2w^2}{3}$

1. Differentiate $A(w)$ with respect to $w$: $A'(w) = \frac{600 - 4w}{3}$

2. Set $A'(w) = 0$ to find the critical points: $600 - 4w = 0 \quad \implies \quad w = 150$

3. Verify that $w = 150$ gives a maximum by examining the second derivative: $A''(w) = \frac{-4}{3} \quad \text{(negative, so it’s a maximum).}$

4. Find $l$ when $w = 150$: $l = \frac{600 - 2(150)}{3} = \frac{600 - 300}{3} = 100$

The dimensions are: $\text{Width: } w = 150 \, \text{feet, Length: } l = 100 \, \text{feet.}$

5. The maximum possible area is: $A = w \cdot l = 150 \cdot 100 = 15,000 \, \text{square feet.}$

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Final Answers:

5. $A(w) = \frac{600w - 2w^2}{3}$

6. Dimensions: $150 \, \text{ft (width)} \times 100 \, \text{ft (length)}$, Maximum Area: $15,000 \, \text{sq ft.}$

Do you want detailed explanations or clarifications on any step?

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5 Follow-up Questions:

1. What does the second derivative test tell us about a function's concavity?
2. How would the solution change if the total fencing were increased or decreased?
3. Could you determine the dimensions graphically instead of using calculus?
4. How do you confirm that the area formula is correct for different values of $w$?
5. What is the significance of the critical point in this problem?

Tip: When optimizing, always verify constraints to ensure feasible results!