Math Problem Statement

A triangle has height = 6y and base = 3y - 1. A rectangle has height = 5y - 7 and base = y + 4 .

Which of the following expressions represents the difference, in terms of y, between the area of the triangle and the area of the rectangle?

13y2 - 19y + 28

4y2 - 16y + 28

3y + 2

14y2 + 10y - 28

Solution

We need to find the difference in terms of yy between the area of the triangle and the area of the rectangle. Let's break it down step by step.

1. Area of the Triangle:

The area AtriangleA_{\text{triangle}} of a triangle is given by the formula:

Atriangle=12×base×heightA_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}

For the triangle, the base is 3y13y - 1 and the height is 6y6y. So, the area of the triangle is:

Atriangle=12×(3y1)×6yA_{\text{triangle}} = \frac{1}{2} \times (3y - 1) \times 6y

Simplifying:

Atriangle=12×(18y26y)=9y23yA_{\text{triangle}} = \frac{1}{2} \times (18y^2 - 6y) = 9y^2 - 3y

2. Area of the Rectangle:

The area ArectangleA_{\text{rectangle}} of a rectangle is given by the formula:

Arectangle=base×heightA_{\text{rectangle}} = \text{base} \times \text{height}

For the rectangle, the base is y+4y + 4 and the height is 5y75y - 7. So, the area of the rectangle is:

Arectangle=(y+4)×(5y7)A_{\text{rectangle}} = (y + 4) \times (5y - 7)

Expanding this:

Arectangle=y(5y7)+4(5y7)=5y27y+20y28=5y2+13y28A_{\text{rectangle}} = y(5y - 7) + 4(5y - 7) = 5y^2 - 7y + 20y - 28 = 5y^2 + 13y - 28

3. Difference Between the Areas:

We now subtract the area of the rectangle from the area of the triangle:

Difference=AtriangleArectangle=(9y23y)(5y2+13y28)\text{Difference} = A_{\text{triangle}} - A_{\text{rectangle}} = (9y^2 - 3y) - (5y^2 + 13y - 28)

Simplifying:

Difference=9y23y5y213y+28=(9y25y2)+(3y13y)+28=4y216y+28\text{Difference} = 9y^2 - 3y - 5y^2 - 13y + 28 = (9y^2 - 5y^2) + (-3y - 13y) + 28 = 4y^2 - 16y + 28

Conclusion:

The expression that represents the difference is:

4y216y+28\boxed{4y^2 - 16y + 28}


Do you want more details or have any questions?

Here are 5 related questions to explore:

  1. What is the area of the triangle when y=2y = 2?
  2. What is the area of the rectangle when y=3y = 3?
  3. How does the difference in area change as yy increases?
  4. How would the formula change if the triangle had a different height?
  5. What is the maximum area of the rectangle as yy varies?

Tip: When solving for areas, always pay attention to the units involved, especially when dealing with different shapes!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Polynomial Operations

Formulas

Area of a triangle: (1/2) × base × height
Area of a rectangle: base × height
Difference of expressions: A - B

Theorems

Distributive Property
Combining Like Terms

Suitable Grade Level

Grades 8-10