An astronaut on the moon throws a baseball upward. The astronaut is 6 fee, 6 inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function
s(t)= -2.7t^2 + 30t + 6.5
The ball will never reach a height of 100 feet. How can this be determined analytically?
After how many seconds is the ball 12 feet above the moon’s surface?
I need help starting off with both
first part:
replace s(t) with 100
100 = -2.7t^2 + 30t + 6.5
2.7t^2 - 30t + 93.5 = 0
This quadratic has no real solution, the discriminant
b^2 - 4ac or (30^2 - 4(2.7)(93.5) is negative.
So it can never reach 100
Does it reach 12 ??
Repeat the above by replacing s(t) with 12
I would use the quadratic equation and use the positive result to find the value of t, the time this happens
In the first part where did you get the 93.5 from
Im not understanding the second part
To determine analytically whether the ball reaches a height of 100 feet, we need to set up an inequality using the given function.
The height of the ball at any given time t is represented by the function s(t) = -2.7t^2 + 30t + 6.5. We need to check if the height of the ball, s(t), ever exceeds 100 feet.
We can set up the inequality: -2.7t^2 + 30t + 6.5 < 100.
Next, we solve this inequality to find the range of values for t that satisfy the condition. Remember that the solutions will be in terms of t.
To find the value of t when the ball is 12 feet above the moon's surface, we need to set s(t) = 12 and solve for t.
Now let's work on solving both problems step by step.
1. Determining whether the ball will reach a height of 100 feet:
Starting with -2.7t^2 + 30t + 6.5 < 100, we rearrange the equation to bring all terms to one side:
-2.7t^2 + 30t + 6.5 - 100 < 0
-2.7t^2 + 30t - 93.5 < 0
To solve this inequality, we can factorize or use the quadratic formula. Factoring may not be easy, so let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
Here, a = -2.7, b = 30, and c = -93.5.
Plugging these values into the formula:
t = (-30 ± √(30^2 - 4(-2.7)(-93.5)))/(2(-2.7))
Simplifying this equation will give two values for t, t1 and t2.
Now, we need to check if these values satisfy the given inequality.
2. Finding when the ball is 12 feet above the moon's surface:
We set s(t) = 12 in the given function:
-2.7t^2 + 30t + 6.5 = 12
Rearranging the equation:
-2.7t^2 + 30t + 6.5 - 12 = 0
-2.7t^2 + 30t - 5.5 = 0
We can now solve this equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = -2.7, b = 30, and c = -5.5.
Plugging these values into the formula:
t = (-30 ± √(30^2 - 4(-2.7)(-5.5)))/(2(-2.7))
Please note that simplifying these equations may result in complex or irrational solutions.