A physics student stands at the top of a hill that has an elevation of 37 meters. He throws a rock and it goes up into the air and then falls back past him and lands on the ground below. The path of the rock can be modeled by the equation y = -0.02x^2 + 0.8x + 37, 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 its starting point will the rock land? Round your answer to the nearest hundredth.
A. 37.00 m***
B. 67.43 m
C. 27.43 m
D. 37.78 m
How many real number solutions does the equation have???
0 = 5x^2 + 2x - 12
A. One solution.
B. Two solutions.
C. Infinitely many solutions.***
D. No solutions.
How many real number solutions does the equation have?
-8x^2 - 8x - 2 = 0
A. One solution.
B. Two solutions.
C. No Solutions.****
D. Infinitely many solutions.
1. X = (-B +- Sqrt(B^2-4AC))/2A.
X = (-0.8 +- sqrt(0.64+2.96))/-0.04 = -27.43, and 67.43. Solution: X = 67.43.
2. 0 = 5x^2+2x-12.
B^2 - 4AC = 4 - (-240) = 244. Two solutions.
3. -8x^2 - 8x - 2 = 0.
B^2-4AC = 64 - 64 = 0. One solution.
When B^2-4AC < 0, No real solutions.
When B^2-4AC = 0, One real solution.
When B^2-4AC > 0, Two real solutions.
Thanks Henry. Really appreciate it bro.
To find the answer to the first question, we need to determine the horizontal distance at which the rock lands. We can find this by setting the equation y = 0 and solving for x.
Given equation: y = -0.02x^2 + 0.8x + 37
Setting y to 0:
0 = -0.02x^2 + 0.8x + 37
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.02, b = 0.8, and c = 37. Plugging these values into the formula, we get:
x = (-(0.8) ± √((0.8)^2 - 4(-0.02)(37))) / (2(-0.02))
Simplifying further, we get:
x = (-0.8 ± √(0.64 + 2.96)) / (-0.04)
x = (-0.8 ± √(3.6)) / (-0.04)
x = (-0.8 ± 1.8974) / (-0.04)
Using the ± and solving for both possibilities:
x₁ = (-0.8 + 1.8974) / (-0.04) ≈ -29.485
x₂ = (-0.8 - 1.8974) / (-0.04) ≈ 68.485
Since we're interested in the positive value for the horizontal distance, we can conclude that the rock lands approximately 68.49 meters horizontally from its starting point.
Therefore, the correct answer is B. 67.43 m.
For the second and third questions, the number of real solutions can be determined by examining the discriminant of the quadratic equation.
The discriminant is given by b^2 - 4ac. If the discriminant is greater than zero, there are two real solutions; if it equals zero, there is one real solution; and if it is less than zero, there are no real solutions.
For the equation 0 = 5x^2 + 2x - 12, the coefficients are a = 5, b = 2, and c = -12. Plugging these values into the discriminant formula, we get:
Discriminant = (2)^2 - 4(5)(-12)
= 4 + 240
= 244
Since the discriminant is positive, there are two real solutions.
Therefore, the correct answer is B. Two solutions.
Similarly, for the equation -8x^2 - 8x - 2 = 0, the coefficients are a = -8, b = -8, and c = -2. Calculating the discriminant:
Discriminant = (-8)^2 - 4(-8)(-2)
= 64 - 64
= 0
Since the discriminant is zero, there is one real solution.
Therefore, the correct answer is A. One solution.