Math Problem Statement

The length of one side of a rectangle is 20 meters shorter than five times the width of the other side. Find the dimensions of the rectangle, given that the area is 60 m^2. The correct measures of both length and width in meters from the choices given are: (6, 10), (12.5, 5), (9, 25), (11.35, 5).

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 60m260 \, \text{m}^2.

Define Variables:

Let:

  • ww: Width of the rectangle (in meters)
  • ll: Length of the rectangle (in meters)

Form Equations:

  1. The length is related to the width by: l=5w20l = 5w - 20

  2. The area of the rectangle is given by: lw=60l \cdot w = 60

Substitute l=5w20l = 5w - 20 into the area equation: (5w20)w=60(5w - 20) \cdot w = 60

Solve for ww:

Expand and simplify: 5w220w=605w^2 - 20w = 60

Rearrange into standard quadratic form: 5w220w60=05w^2 - 20w - 60 = 0

Divide the entire equation by 5 to simplify: w24w12=0w^2 - 4w - 12 = 0

Factorize the quadratic equation: (w6)(w+2)=0(w - 6)(w + 2) = 0

Solve for ww: w=6orw=2w = 6 \quad \text{or} \quad w = -2

Since width cannot be negative: w=6metersw = 6 \, \text{meters}

Find ll:

Substitute w=6w = 6 into l=5w20l = 5w - 20: l=5(6)20=3020=10metersl = 5(6) - 20 = 30 - 20 = 10 \, \text{meters}

Verify:

Check the area: lw=106=60m2l \cdot w = 10 \cdot 6 = 60 \, \text{m}^2

Final Answer:

The dimensions of the rectangle are:

  • Width: 6meters6 \, \text{meters}
  • Length: 10meters10 \, \text{meters}

Correct Option:

The answer is 6,106, 10.


Would you like further clarification or more examples?

Related Questions:

  1. How do you solve other quadratic equations geometrically?
  2. What are common applications of area equations in real life?
  3. How would this problem change if the rectangle was a square?
  4. Can you solve similar problems using graphing methods?
  5. 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