To determine the highest point the shot put reaches, we need to find the vertex of the parabola represented by the equation
[tex]\[ y = -0.01x^2 + 0.7x + 6. \][/tex]
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] is given by the coordinates [tex]\((x, y)\)[/tex], where the x-coordinate can be found using the formula:
[tex]\[ x = -\frac{b}{2a}. \][/tex]
First, we identify the coefficients from the equation:
[tex]\[ a = -0.01, \][/tex]
[tex]\[ b = 0.7, \][/tex]
[tex]\[ c = 6. \][/tex]
Plugging in these values into the formula for the x-coordinate of the vertex, we get:
[tex]\[ x = -\frac{0.7}{2 \times -0.01} = -\frac{0.7}{-0.02} = 35. \][/tex]
Next, we substitute this x value back into the original equation to find the corresponding y value, which represents the highest point:
[tex]\[ y = -0.01(35)^2 + 0.7(35) + 6. \][/tex]
Calculating each term, we have:
[tex]\[ -0.01(35)^2 = -0.01 \times 1225 = -12.25, \][/tex]
[tex]\[ 0.7(35) = 24.5, \][/tex]
[tex]\[ 6 = 6. \][/tex]
Adding these together:
[tex]\[ y = -12.25 + 24.5 + 6 = 18.25. \][/tex]
Therefore, the highest point the shot put reaches is at the coordinates:
[tex]\[ (35, 18.25). \][/tex]
So, the highest point is 18.25 feet when the horizontal distance is 35 feet.
