Let's define: - \( w \) = **front width** (in inches) - \( d \) = **depth** (in inches, the side perpendicular to front width) - \( h \) = **height** (in inches) Given: - **Volume:** \( wdh = 250 \) - **Base cost:** 8 cents/in² - **Front cost:** 12 cents/in² - **Other sides cost:** 2 cents/in² ### 1. **Write the cost function** #### **Areas:** - **Base (bottom):** \( wd \) - **Front (decorated):** \( wh \) - **Back:** \( wh \) - **Sides (left and right):** \( dh \) each #### **Cost:** - **Base:** \( 8(wd) \) - **Front:** \( 12(wh) \) - **Back:** \( 2(wh) \) - **Sides:** \( 2(dh) \cdot 2 = 4dh \) **Total Cost:** \[ C = 8wd + 12wh + 2wh + 4dh \] \[ C = 8wd + 14wh + 4dh \] ### 2. **Express one variable in terms of the others using the volume constraint:** \[ wdh = 250 \implies d = \frac{250}{wh} \] ### 3. **Substitute \( d \) in the cost function:** \[ C(w, h) = 8w\left(\frac{250}{wh}\right) + 14wh + 4\left(\frac{250}{wh}\right)h \] \[ C(w, h) = \frac{2000}{h} + 14wh + \frac{1000}{w} \] ### 4. **Minimize \( C(w, h) \) with respect to \( w \) and \( h \):** Take the partial derivatives and set to zero. #### **First: with respect to \( w \)** \[ \frac{\partial C}{\partial w} = 14h - \frac{1000}{w^2} = 0 \] \[ 14h = \frac{1000}{w^2} \] \[ w^2 = \frac{1000}{14h} \] \[ w = \sqrt{\frac{1000}{14h}} \] #### **Second: with respect to \( h \)** \[ \frac{\partial C}{\partial h} = -\frac{2000}{h^2} + 14w = 0 \] \[ 14w = \frac{2000}{h^2} \] \[ w = \frac{2000}{14h^2} \] Set the two expressions for \( w \) equal: \[ \sqrt{\frac{1000}{14h}} = \frac{2000}{14h^2} \] Square both sides: \[ \frac{1000}{14h} = \left(\frac{2000}{14h^2}\right)^2 \] \[ \frac{1000}{14h} = \frac{4{,}000{,}000}{196h^4} \] Multiply both sides by \( 196h^4 \): \[ 1000 \cdot 196h^4 = 14h \cdot 4{,}000{,}000 \] \[ 196{,}000 h^4 = 56{,}000{,}000 h \] \[ h^3 = \frac{56{,}000{,}000}{196{,}000} \] \[ h^3 = 285.714... \] \[ h = \sqrt[3]{285.714...} \] \[ h \approx 6.6 \text{ in.} \] ### 5. **Find \( w \) and \( d \):** #### \( w = \sqrt{\frac{1000}{14h}} \) \[ w = \sqrt{\frac{1000}{14 \times 6.6}} \] \[ w = \sqrt{\frac{1000}{92.4}} \] \[ w = \sqrt{10.82} \] \[ w \approx 3.29 \text{ in.} \] #### \( d = \frac{250}{wh} \) \[ d = \frac{250}{3.29 \times 6.6} \] \[ d = \frac{250}{21.714} \] \[ d \approx 11.51 \text{ in.} \] --- ## **Final Answers:** - **Front width:** \(\boxed{3.29} \) in. - **Depth:** \(\boxed{11.51} \) in. - **Height:** \(\boxed{6.6} \) in. Let me know if you need the cost calculation or a check!

Certainly! Here's a detailed, step-by-step solution with all calculations and explanations: --- ### **Problem Restatement:** Construct an open-top rectangular box with volume \( 250\, \text{in}^3 \). Cost details: - Base material costs **8 cents/in²** - Front decorated costs **12 cents/in²** - Other sides cost **2 cents/in²** **Find the dimensions (front width \(w\), depth \(d\), height \(h\)) that minimize total construction cost.** --- ### **Step 1: Define Variables** - \( w \) = **front width** (in inches) - \( d \) = **depth** (in inches) - \( h \) = **height** (in inches) ### **Step 2: Write the volume constraint** \[ w \times d \times h = 250 \] \[ \Rightarrow d = \frac{250}{wh} \] ### **Step 3: Write the cost function** **Areas and their costs:** | Part | Area | Cost per in² | Total Cost Contribution | |---------|------------------------------|------------------------|----------------------------| | Base | \( wd \) | 8 cents | \( 8 \times wd \) | | Front (decorated) | \( wh \) | 12 cents | \( 12 \times wh \) | | Back | \( wh \) | 2 cents | \( 2 \times wh \) | | Two sides (left and right) | \( 2 \times dh \) | 2 cents for each side | \( 2 \times dh \times 2 = 4dh \) | **Total cost \( C \):** \[ C = 8wd + 12wh + 2wh + 4dh = 8wd + 14wh + 4dh \] --- ### **Step 4: Substitute \( d \) into the cost function** \[ d = \frac{250}{wh} \] Plug into \( C \): \[ C(w, h) = 8w \left( \frac{250}{wh} \right) + 14wh + 4 \left( \frac{250}{wh} \right) h \] Simplify each term: - First term: \[ 8w \times \frac{250}{wh} = \frac{8 \times 250}{h} = \frac{2000}{h} \] - Last term: \[ 4 \times \frac{250}{wh} \times h = 4 \times \frac{250h}{wh} = 4 \times \frac{250}{w} = \frac{1000}{w} \] Now, the cost function simplifies to: \[ \boxed{ C(w, h) = \frac{2000}{h} + 14wh + \frac{1000}{w} } \] --- ### **Step 5: Minimize \( C(w, h) \) using calculus** We find critical points by setting partial derivatives to zero. --- ### **Part A: Partial derivative with respect to \( w \)** \[ \frac{\partial C}{\partial w} = 14h - \frac{1000}{w^2} = 0 \] Solve for \( w \): \[ 14h = \frac{1000}{w^2} \] \[ w^2 = \frac{1000}{14h} \] \[ \Rightarrow w = \sqrt{\frac{1000}{14h}} \quad \text{(Equation 1)} \] --- ### **Part B: Partial derivative with respect to \( h \)** \[ \frac{\partial C}{\partial h} = -\frac{2000}{h^2} + 14w = 0 \] \[ 14w = \frac{2000}{h^2} \] \[ w = \frac{2000}{14h^2} \quad \text{(Equation 2)} \] --- ### **Step 6: Equate equations for \( w \)** From Equations 1 and 2: \[ \sqrt{\frac{1000}{14h}} = \frac{2000}{14h^2} \] --- ### **Step 7: Solve for \( h \)** Square both sides: \[ \left( \sqrt{\frac{1000}{14h}} \right)^2 = \left( \frac{2000}{14h^2} \right)^2 \] \[ \frac{1000}{14h} = \frac{4,000,000}{196 h^4} \] Multiply both sides by \( 196 h^4 \): \[ \frac{1000}{14h} \times 196 h^4 = 4,000,000 \] Simplify the left side: \[ 1000 \times 14 h^4 = 4,000,000 \] Because: \[ \frac{196}{14} = 14 \] So: \[ 1000 \times 14 h^4 = 4,000,000 \] \[ 14,000 h^4 = 4,000,000 \] Divide both sides by 14,000: \[ h^4 = \frac{4,000,000}{14,000} = \frac{4000000}{14000} \approx 285.714 \] Find \( h \): \[ h = \sqrt[4]{285.714} \] Calculate: \[ \sqrt{285.714} \approx 16.91 \] Then: \[ h = \sqrt{16.91} \approx 4.11 \] **Note:** Our earlier calculation approximated \( h \) as about 6.6, but this refined calculation suggests approximately **4.11 inches**. Let's double-check: --- ### **Alternative approach: precise calculation** From previous step: \[ h^4 = 285.714 \] \[ h = (285.714)^{1/4} \] Using a calculator: \[ h \approx e^{\frac{1}{4} \ln 285.714} \] \[ \ln 285.714 \approx 5.652 \] \[ h \approx e^{5.652/4} = e^{1.413} \approx 4.11 \] **Final \( h \approx 4.11 \) inches.** --- ### **Step 8: Find \( w \)** Using Equation 2: \[ w = \frac{2000}{14 h^2} \] Calculate \( h^2 \): \[ h^2 = (4.11)^2 \approx 16.89 \] Now: \[ w = \frac{2000}{14 \times 16.89} = \frac{2000}{236.46} \approx 8.45 \] --- ### **Step 9: Find \( d \)** Recall: \[ d = \frac{250}{w h} \] Compute: \[ w h = 8.45 \times 4.11 \approx 34.75 \] \[ d = \frac{250}{34.75} \approx 7.20 \] --- ### **Final Dimensions:** | Dimension | Value (in inches) | |------------|-------------------| | **Front width \(w\)** | \(\boxed{8.45}\) in | | **Height \(h\)** | \(\boxed{4.11}\) in | | **Depth \(d\)** | \(\boxed{7.20}\) in | --- ### **Summary:** - **Front width:** **8.45 inches** - **Height:** **4.11 inches** - **Depth:** **7.20 inches** These dimensions minimize the total cost of constructing the box with the given constraints. --- **Let me know if you'd like me to verify the total cost or provide further explanations!**