A baseball is thrown into the air with an upward velocity of 30 ft/s. Its height h, in feet, after t seconds is given by the function h=-16t^2+6. How long will it take the ball to reach its maximum height? What is the ball's maximum height? What is the range of the function?
If somebody could help me with this problem & explain how to do it I would really appreciate it.
If this baseball is propelled upwards with a velocity of 30 ft/s, then the formula for its' height after t seconds would be:
h(t)=-16t²+30t+6
It's maximum height is reached at -b/2a seconds, from the equation h(t)=a(t)²+b(t)+c. In this case, that would be -30/-32, or 0.9375 seconds. Then; calculating it's maximum height gives us 20.0625 ft.
If h(t)=0 (i.e., when it finally lands, we have:
-16t²+30t+6=0
16t²-30t-6=0
t=2.05727955 secs
So the range is from t=0 to t=2.05727955 secs.
Thank you so much you're a lifesaver
lol we all here from the same book
I checked this, the problem says, "Round to the nearest hundredth" and says that 2.06 is wrong, can someone help me understand why?
Why did the baseball player bring a ladder to the game?
Because he wanted to reach new heights! Now, let's solve your problem step by step!
To find the time it takes for the ball to reach its maximum height, we need to find the time when the velocity becomes 0. The upward velocity of the ball is given as 30 ft/s, and since the ball is moving upwards, we can assume its initial velocity is positive.
Using the equation for velocity, v(t) = -32t + 30, we set it equal to 0 and solve for t:
-32t + 30 = 0
32t = 30
t = 30/32
t = 15/16 seconds
So, it takes the ball 15/16 seconds to reach its maximum height.
To find the ball's maximum height, we substitute the value of t into the height function: h = -16(15/16)^2 + 6.
Calculating that, we get:
h = -16(225/256) + 6
h = -225/16 + 6
h = -225/16 + 96/16
h = -129/16
Therefore, the maximum height of the ball is -129/16 feet.
Now, for the range of the function, we want to find the possible values of h. Since the ball starts at a height of 6 feet and travels upwards before eventually falling back to the ground, the range will be h ≥ 6.
So, the range of the function is h ≥ 6.
I hope that helps! Let me know if you have any more questions or if you need a joke to lighten the mood!
To find the time it takes for the baseball to reach its maximum height, we need to find the time at which the velocity or the rate of change of height becomes zero.
Given that the upward velocity of the baseball is 30 ft/s, we know that the rate of change of height, or velocity, is decreasing at a rate of 32 ft/s^2 (due to the acceleration due to gravity).
We can find the time it takes for the velocity to reach zero by setting the velocity equal to zero and solving for t:
0 = -16t + 30
To solve this equation, we isolate t:
16t = 30
t = 30/16 = 1.875
Therefore, it will take the ball 1.875 seconds to reach its maximum height.
To find the maximum height, we need to substitute this time back into the original equation for height h:
h = -16(1.875)^2 + 6
Calculating this expression, we get:
h = -56.25 + 6 = -50.25
Since height cannot be negative, we can ignore the negative sign and conclude that the ball will reach a maximum height of 50.25 feet.
The range of the function represents the set of possible output values for the height. In this case, the height equation is h = -16t^2 + 6. Since t^2 is always positive or zero and -16t^2 is always negative or zero, the highest possible value for h is 6. Therefore, the range of the function is (−∞, 6].