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Math  /  Algebra

Question18. Kate is an industrial design engineer. She is creating the program for cutting fabric for a shade sail. The shape of a shade sail is defined by three intersecting parabolas. The equations of the parabolas are y=x2+8x+16y=x2−8x+16y=−x28+2\begin{array}{l} y=x^{2}+8 x+16 \\ y=x^{2}-8 x+16 \\ y=-\frac{x^{2}}{8}+2 \end{array} where xx and yy are measurements in metres. a) Use an algebraic method to determine the coordinates of the three vertices of the sail. b) Estimate the

Studdy Solution

STEP 1

1. The problem involves three parabolas defined by given equations.
2. The task is to find the vertices of these parabolas.
3. The equations are in standard form for parabolas.
4. The coordinates of the vertices are needed for part (a).

STEP 2

1. Identify the vertex of the first parabola.
2. Identify the vertex of the second parabola.
3. Identify the vertex of the third parabola.

STEP 3

Identify the vertex of the first parabola.
The equation is y=x2+8x+16 y = x^2 + 8x + 16 .
To find the vertex, use the formula for the vertex of a parabola y=ax2+bx+c y = ax^2 + bx + c , which is x=−b2a x = -\frac{b}{2a} .
For this equation, a=1 a = 1 , b=8 b = 8 , and c=16 c = 16 .
Calculate x x :
x=−82×1=−4 x = -\frac{8}{2 \times 1} = -4
Substitute x=−4 x = -4 back into the equation to find y y :
y=(−4)2+8(−4)+16 y = (-4)^2 + 8(-4) + 16 y=16−32+16 y = 16 - 32 + 16 y=0 y = 0
The vertex of the first parabola is (−4,0)(-4, 0).

STEP 4

Identify the vertex of the second parabola.
The equation is y=x2−8x+16 y = x^2 - 8x + 16 .
Use the vertex formula x=−b2a x = -\frac{b}{2a} .
For this equation, a=1 a = 1 , b=−8 b = -8 , and c=16 c = 16 .
Calculate x x :
x=−−82×1=4 x = -\frac{-8}{2 \times 1} = 4
Substitute x=4 x = 4 back into the equation to find y y :
y=(4)2−8(4)+16 y = (4)^2 - 8(4) + 16 y=16−32+16 y = 16 - 32 + 16 y=0 y = 0
The vertex of the second parabola is (4,0)(4, 0).

STEP 5

Identify the vertex of the third parabola.
The equation is y=−x28+2 y = -\frac{x^2}{8} + 2 .
Use the vertex formula x=−b2a x = -\frac{b}{2a} .
For this equation, a=−18 a = -\frac{1}{8} , b=0 b = 0 , and c=2 c = 2 .
Calculate x x :
x=−02×−18=0 x = -\frac{0}{2 \times -\frac{1}{8}} = 0
Substitute x=0 x = 0 back into the equation to find y y :
y=−(0)28+2 y = -\frac{(0)^2}{8} + 2 y=2 y = 2
The vertex of the third parabola is (0,2)(0, 2).
The vertices of the three parabolas are:
1. (−4,0)(-4, 0)
2. (4,0)(4, 0)
3. (0,2)(0, 2)

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