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## Sagot :

### 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.