Math Problem Statement
The frame on a picture is 8 in by 10 in outside and is of uniform width. What is the width of the frame if 48 in^2 of the picture shows?
Solution
Let's solve the problem step by step.
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Define the variables:
- Let be the width of the frame.
- The outside dimensions of the frame are 8 inches by 10 inches, so the picture inside the frame will be smaller due to the width of the frame.
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Calculate the dimensions of the picture showing:
- The actual width of the picture (inside the frame) will be inches (since the frame reduces the width on both sides by ).
- The actual height of the picture (inside the frame) will be inches (since the frame reduces the height on both sides by ).
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Set up the area equation:
- The area of the picture showing is given as 48 square inches. The area of the picture is the product of its width and height, so:
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Expand and simplify the equation: Therefore, the equation becomes: Simplifying further: Divide the entire equation by 4:
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Solve the quadratic equation: We can factor this equation: So, the possible solutions for are:
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Interpret the solution: Since the frame width must be smaller than the overall dimensions of the frame, is not possible (the frame would completely cover the picture), so the only feasible solution is .
Thus, the width of the frame is 1 inch.
Would you like more details on any part of this solution?
Relative Questions:
- How do you solve quadratic equations by factoring?
- What are some alternative methods to solve quadratic equations?
- How does changing the dimensions of the frame affect the area of the picture?
- Can the quadratic formula always be used for problems like this one?
- What happens if the picture frame's area is less than 48 square inches?
Tip: Always check the physical feasibility of solutions after solving mathematical equations!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Geometry (Area Calculation)
Formulas
Area = width × height
Quadratic equation: ax^2 + bx + c = 0
Theorems
Quadratic Formula
Factoring of Quadratics
Suitable Grade Level
Grades 8-10
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