To find the overall dimensions of the poster that uses the least amount of paper, we need to calculate the width and height of the poster based on the given requirements.

Step-by-step solution:

1. The photo area:
   The photo is 98 in². Let the width of the photo be $x$ and the height of the photo be $y$, such that:
   $x \cdot y = 98$

2. The border dimensions:

   • The poster has a 2 in border at the top and bottom, which adds $2 + 2 = 4$ inches to the height.
   • The poster has a 1 in border on each side, which adds $1 + 1 = 2$ inches to the width.

   So, the total dimensions of the poster become: $\text{Width of poster} = x + 2$ $\text{Height of poster} = y + 4$

3. Goal: Minimize the total area of the poster:
   The total area of the poster is given by: $A = (\text{Width of poster}) \times (\text{Height of poster}) = (x + 2)(y + 4)$

   Substituting $y = \frac{98}{x}$ into the equation (from the photo area $x \cdot y = 98$): $A = (x + 2) \left( \frac{98}{x} + 4 \right)$

   Simplify the expression: $A = (x + 2) \left( \frac{98}{x} \right) + (x + 2)(4)$ $A = 98 + \frac{196}{x} + 4x + 8$ $A = 106 + \frac{196}{x} + 4x$

4. Minimize $A$: To find the value of $x$ that minimizes $A$, we take the derivative of $A$ with respect to $x$ and set it to zero: $\frac{dA}{dx} = -\frac{196}{x^2} + 4$ Set $\frac{dA}{dx} = 0$: $-\frac{196}{x^2} + 4 = 0$ $\frac{196}{x^2} = 4$ $x^2 = \frac{196}{4} = 49$ $x = \sqrt{49} = 7$

5. Find $y$:
   Using $x \cdot y = 98$, we find: $y = \frac{98}{7} = 14$

6. Determine the overall poster dimensions:
   Substituting $x = 7$ and $y = 14$ into the border dimensions:

   • Width of poster = $x + 2 = 7 + 2 = 9$ inches
   • Height of poster = $y + 4 = 14 + 4 = 18$ inches

Final Answer:

The overall dimensions of the poster should be 9 inches by 18 inches to use the least amount of paper.

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Would you like further explanation of any steps?

Related Questions:

1. Why does minimizing $A$ involve taking the derivative?
2. What would the total area of the poster be with these dimensions?
3. How does the size of the border affect the solution?
4. Could we change the aspect ratio of the photo to further minimize paper usage?
5. What if the photo area was a different size, e.g., 120 in²—how would the dimensions change?

Tip: Always check constraints (like the border dimensions) when optimizing problems involving geometry!