A physics student stands on top of a hill that has an elevation of 56 meters. He throws a rock and it goes up into the air and then falls past him and lands on the ground below. The path of the rock can be modeled by the equation -

y=-0.04x^2+1.3x+56 where x is the horizontal distance in meters from the starting point on the top of the hill and y is the height in meters of the rock above the ground.
How far horizontally from the starting point will the rock land?

a- 56.0m
b- 24.54m
c- 57.,04m
d- 57.26m

Grandpa asking for help again.

thank you

To find the horizontal distance the rock will land, we need to find the value of x when y equals 0.

Given the equation: y = -0.04x^2 + 1.3x + 56

Setting y = 0, we can solve the equation for x:

0 = -0.04x^2 + 1.3x + 56

This quadratic equation can be solved using factoring or the quadratic formula. Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = -0.04, b = 1.3, and c = 56.

Calculating the discriminant: b^2 - 4ac, we get:

b^2 - 4ac = (1.3)^2 - 4(-0.04)(56) = 1.69 + 8.96 = 10.65

Now we can substitute the values into the quadratic formula:

x = (-1.3 ± √(10.65)) / (2*(-0.04))

Calculating this expression, we get two possible values for x:

x = (-1.3 + √(10.65)) / (-0.08) ≈ 24.54

x = (-1.3 - √(10.65)) / (-0.08) ≈ 57.26

Therefore, the rock will land approximately 24.54 meters horizontally from the starting point.

So, the correct answer is option b) 24.54m.

To find out how far horizontally from the starting point the rock will land, we need to determine the value of x when the height y is equal to zero. This is because when the rock lands on the ground, its height will be zero.

Given the equation of the path of the rock: y = -0.04x^2 + 1.3x + 56, we can substitute y with zero:

0 = -0.04x^2 + 1.3x + 56

Now, to solve for x, we can use the quadratic formula:

x = (-b ± √(b^2-4ac)) / (2a)

In this case, a = -0.04, b = 1.3, and c = 56. Plugging these values into the quadratic formula, we get:

x = (-1.3 ± √(1.3^2 - 4*(-0.04)*56)) / (2*(-0.04))

Simplifying further:

x = (-1.3 ± √(1.69 + 8.96)) / (-0.08)

x = (-1.3 ± √10.65) / -0.08

Now, calculating both possible solutions:

x₁ = (-1.3 + √10.65) / -0.08 ≈ -57.04
x₂ = (-1.3 - √10.65) / -0.08 ≈ 57.26

Since distance cannot be negative, we discard the negative solution, x₁ ≈ -57.04, and conclude that the rock will land approximately 57.26 meters horizontally from the starting point.

Therefore, the correct option is d) 57.26m.

0 = -0.04x^2+1.3x+56

or
0 = 0.04x^2 - 1.3x - 56
solve quadratic