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

We need to find the difference in terms of \( y \) 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 \( A_{\text{triangle}} \) of a triangle is given by the formula:

\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]

For the triangle, the base is \( 3y - 1 \) and the height is \( 6y \). So, the area of the triangle is:

\[
A_{\text{triangle}} = \frac{1}{2} \times (3y - 1) \times 6y
\]

Simplifying:

\[
A_{\text{triangle}} = \frac{1}{2} \times (18y^2 - 6y) = 9y^2 - 3y
\]

### 2. Area of the Rectangle:
The area \( A_{\text{rectangle}} \) of a rectangle is given by the formula:

\[
A_{\text{rectangle}} = \text{base} \times \text{height}
\]

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

\[
A_{\text{rectangle}} = (y + 4) \times (5y - 7)
\]

Expanding this:

\[
A_{\text{rectangle}} = y(5y - 7) + 4(5y - 7) = 5y^2 - 7y + 20y - 28 = 5y^2 + 13y - 28
\]

### 3. Difference Between the Areas:
We now subtract the area of the rectangle from the area of the triangle:

\[
\text{Difference} = A_{\text{triangle}} - A_{\text{rectangle}} = (9y^2 - 3y) - (5y^2 + 13y - 28)
\]

Simplifying:

\[
\text{Difference} = 9y^2 - 3y - 5y^2 - 13y + 28 = (9y^2 - 5y^2) + (-3y - 13y) + 28 = 4y^2 - 16y + 28
\]

### Conclusion:
The expression that represents the difference is:

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

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

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

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?

13y^2 - 19y + 28
4y^2 - 16y + 28
3y + 2
14y^2 + 10y - 28

Learn how to determine the difference between the areas of a triangle and a rectangle, both expressed in terms of y. This detailed solution covers essential algebraic and geometric concepts suitable for students in Grades 8-10.

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