Question

numerical response #2
the height of a golf ball, h(d), in metres, is modelled by the function ( h(d) = -0.20d^2 + 1.94d ), where d is the horizontal distance the ball has traveled, in metres, after being struck. the horizontal distance the ball has traveled, to the nearest tenth of a metre, when it is at its maximum height, is
(record your answer on the answer sheet.)

numerical response #3
the solution to the quadratic equation ( 11x^2 - 8x - 1 = 0 ) can be written in the form ( x = \frac{e pm fsqrt{g}}{h} ). when x is simplified, the sum of ( e + f + g + h ) is ______.
(record your answer on the answer sheet.)

1. the graph of the inequality ( 9x - 3y > 11 ) would show

a. a dotted boundary line with shading above the line
b. a solid boundary line with shading below the line
c. a dotted boundary line with shading below the line
d. a solid boundary line with shading above the line

Explanation:

Numerical Response #2

Step1: Identify the function type

The function \( h(d) = -0.20d^2 + 1.94d \) is a quadratic function in the form \( y = ax^2 + bx + c \), where \( a = -0.20 \), \( b = 1.94 \), and \( c = 0 \). For a quadratic function, the vertex (which gives the maximum or minimum point) occurs at \( d = -\frac{b}{2a} \). Since \( a < 0 \), the parabola opens downward, so the vertex is the maximum point.

Step2: Calculate the horizontal distance at maximum height

Substitute \( a = -0.20 \) and \( b = 1.94 \) into the formula \( d = -\frac{b}{2a} \):

$$ d = -\frac{1.94}{2 \times (-0.20)} = -\frac{1.94}{-0.40} = \frac{1.94}{0.40} = 4.85 $$

Rounding to the nearest tenth of a metre, we get \( 4.9 \).

Step1: Recall the quadratic formula

The quadratic formula for a quadratic equation \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For the equation \( 11x^2 - 8x - 1 = 0 \), we have \( a = 11 \), \( b = -8 \), and \( c = -1 \).

Step2: Identify \( e \), \( f \), \( g \), and \( h \)

Comparing with the form \( x = \frac{e \pm f\sqrt{g}}{h} \):

• \( e = -b = 8 \) (since \( b = -8 \) in the equation, so \( -b = 8 \))
• \( f = 1 \) (coefficient of the square root term)
• \( g = b^2 - 4ac = (-8)^2 - 4(11)(-1) = 64 + 44 = 108 \)
• \( h = 2a = 2 \times 11 = 22 \)

Step3: Calculate the sum \( e + f + g + h \)

$$ e + f + g + h = 8 + 1 + 108 + 22 = 139 $$

1. Boundary Line Type: For the inequality \( 9x - 3y > 11 \), since the inequality is strict (\( > \), not \( \geq \)), the boundary line should be dotted (not solid).
2. Shading Direction: Rewrite the inequality in slope - intercept form (\( y = mx + b \)) to determine the shading.

• Start with \( 9x - 3y > 11 \).
• Subtract \( 9x \) from both sides: \( -3y > -9x + 11 \).
• Divide both sides by \( -3 \). When dividing an inequality by a negative number, the direction of the inequality sign changes: \( y < 3x-\frac{11}{3} \).
• For the inequality \( y < 3x-\frac{11}{3} \), we shade below the line (because \( y \) is less than the expression on the right - hand side).

Answer:

\( 4.9 \)

Numerical Response #3