Math Problem Statement
Solution
Let’s solve this step by step:
Problem Analysis
We are tasked with finding the dimensions of a rectangle where:
- The length is 20 meters shorter than five times the width.
- The area of the rectangle is .
Define Variables:
Let:
- : Width of the rectangle (in meters)
- : Length of the rectangle (in meters)
Form Equations:
-
The length is related to the width by:
-
The area of the rectangle is given by:
Substitute into the area equation:
Solve for :
Expand and simplify:
Rearrange into standard quadratic form:
Divide the entire equation by 5 to simplify:
Factorize the quadratic equation:
Solve for :
Since width cannot be negative:
Find :
Substitute into :
Verify:
Check the area:
Final Answer:
The dimensions of the rectangle are:
- Width:
- Length:
Correct Option:
The answer is .
Would you like further clarification or more examples?
Related Questions:
- How do you solve other quadratic equations geometrically?
- What are common applications of area equations in real life?
- How would this problem change if the rectangle was a square?
- Can you solve similar problems using graphing methods?
- How do quadratic equations apply to physical measurements like volume?
Tip:
Always double-check units in area problems to ensure consistency!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Geometry (Area of Rectangle)
Formulas
Area of a rectangle: Area = Length × Width
Quadratic equation: ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 8-10
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