Let's tackle each part of the question step by step.
### Part (a): Distance the Soccer Ball Travelled Horizontally When It Lands on the Ground
Given the equation of the soccer ball's path:
[tex]\[ h = -0.1d^2 + d + 0.5 \][/tex]
To find the horizontal distance [tex]\(d\)[/tex] when the soccer ball lands on the ground, we need to set the height [tex]\(h\)[/tex] to 0 since the ball will be at 0 meters above the ground when it lands. So, we solve the following equation:
[tex]\[ 0 = -0.1d^2 + d + 0.5 \][/tex]
Solving this quadratic equation for [tex]\(d\)[/tex], we get two solutions. Since distance cannot be negative, we select the positive solution, which is approximately:
[tex]\[ d \approx 10.5 \text{ meters} \][/tex]
So, when the soccer ball lands on the ground, it has traveled approximately 10.5 meters horizontally.
### Part (b): Horizontal Distance When the Soccer Ball is at a Height of 2.6 Meters
Next, we are asked to find the horizontal distance [tex]\(d\)[/tex] when the soccer ball reaches a height [tex]\(h\)[/tex] of 2.6 meters. We use the same path equation and set [tex]\(h\)[/tex] to 2.6:
[tex]\[ 2.6 = -0.1d^2 + d + 0.5 \][/tex]
Rearranging the equation to solve for [tex]\(d\)[/tex]:
[tex]\[ -0.1d^2 + d + 0.5 - 2.6 = 0 \][/tex]
[tex]\[ -0.1d^2 + d - 2.1 = 0 \][/tex]
Solving this quadratic equation for [tex]\(d\)[/tex], we get two solutions. Again, we choose the positive solution:
[tex]\[ d \approx 7.0 \text{ meters} \][/tex]
Therefore, when the soccer ball is at a height of 2.6 meters, it has traveled approximately 7.0 meters horizontally.
### Summary of Results
a) The soccer ball travels approximately 10.5 meters horizontally when it lands on the ground.
b) The horizontal distance of the soccer ball when it is at a height of 2.6 meters is approximately 7.0 meters.
