To determine the area of the triangle formed by the given parabolas, we follow these steps:

Step 1: Find the Intersection Points

The given equations are:

1. $y = -17(x - 7) - 30.3$
2. $y = -17(x - 7) + 15$
3. $y = 18(x - 23.9) - 30.3$

Since the first two equations have the same slope ($-17$), they represent parallel lines, not parabolas. So, these are actually linear equations, and the figure formed will be a triangle.

Solving for intersection points:

Intersection of Equations (1) and (2):

Set the right-hand sides equal to each other:

$-17(x - 7) - 30.3 = -17(x - 7) + 15$

Since the left-hand side cancels out, this simplifies to:

$-30.3 = 15$

This is a contradiction, meaning these two lines are parallel and never intersect.

Thus, we only need to find the intersections of:

• $y = -17(x - 7) - 30.3$ with $y = 18(x - 23.9) - 30.3$
• $y = -17(x - 7) + 15$ with $y = 18(x - 23.9) - 30.3$

Intersection of (1) and (3):

$-17(x - 7) - 30.3 = 18(x - 23.9) - 30.3$

Cancel $-30.3$:

$-17(x - 7) = 18(x - 23.9)$

Expanding:

$-17x + 119 = 18x - 430.2$

$119 + 430.2 = 18x + 17x$

$549.2 = 35x$

$x = 15.6929$

Substituting $x = 15.6929$ into $y = -17(x - 7) - 30.3$:

$y = -17(15.6929 - 7) - 30.3$

$y = -17(8.6929) - 30.3$

$y = -147.7793 - 30.3 = -178.0793$

So, one intersection point is $(15.7, -178.1)$ (rounded to one decimal).

Intersection of (2) and (3):

$-17(x - 7) + 15 = 18(x - 23.9) - 30.3$

$-17x + 119 + 15 = 18x - 430.2$

$134 = 35x - 430.2$

$35x = 564.2$

$x = 16.12$

Substituting $x = 16.12$ into $y = -17(x - 7) + 15$:

$y = -17(16.12 - 7) + 15$

$y = -17(9.12) + 15$

$y = -155.04 + 15 = -140.04$

So, the second intersection point is $(16.1, -140.0)$ (rounded to one decimal).

Intersection of (1) and (2) (Parallel Lines)

These lines never intersect.

Step 2: Use the Triangle Area Formula

The three vertices of the triangle are:

1. $A(15.7, -178.1)$
2. $B(16.1, -140.0)$
3. $C(23.9, -30.3)$ (from the given equation)

Using the formula for the area of a triangle given three points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

$A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substituting:

$A = \frac{1}{2} \left| 15.7(-140 + 30.3) + 16.1(-30.3 + 178.1) + 23.9(-178.1 + 140) \right|$

$A = \frac{1}{2} \left| 15.7(-109.7) + 16.1(147.8) + 23.9(-38.1) \right|$

$A = \frac{1}{2} \left| -1723.29 + 2379.58 - 911.59 \right|$

$A = \frac{1}{2} \left| -255.3 \right|$

$A = \frac{1}{2} \times 255.3 = 127.7$

Final Answer:

The area of the triangle is 127.7 square units (correct to one decimal place).

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Related Questions:

1. How do you determine if three lines form a triangle?
2. What is the formula for the area of a triangle given three vertices?
3. How do you check if two lines are parallel?
4. What is the general form of a linear equation?
5. How do you solve a system of equations graphically?

Tip:

When solving for intersection points, always check for parallel lines first to avoid unnecessary calculations.