Baseball
Nick throws a baseball upward with an initial velocity of 60 feet per second. What is the maximum height of the ball after Nick releases the ball?
Formula for the problem - h(t)=v0t-16t^2, h(t)= the height of an object in feet, v0=an object's initial velocity in feet per second, and t= time in seconds
but you know the initial velocity
so h(t) = 60t - 16t^2
If you are studying calculus, differentiate, set the derivative equal to zero and solve for t
Substitute the t value back in your equation.
If you don't know Calculus, use the method of completing the square, I hope you know what i mean by that.
I don't know what you mean by that
If you're not familiar with calculus, you can still find the maximum height of the ball by using the method of completing the square.
To do this, let's start with the equation h(t) = 60t - 16t^2 that you mentioned.
Step 1: Rewrite the equation in the form h(t) = -16t^2 + 60t.
Step 2: Notice that this equation is in the form h(t) = at^2 + bt, where a = -16 and b = 60.
Step 3: To complete the square, divide b by 2a and square the result. In this case, b/2a = 60/(2*(-16)) = -1.875. Squaring this value, we get (-1.875)^2 = 3.5156.
Step 4: Add this squared value to both sides of the equation to create a perfect square trinomial. The equation becomes h(t) + 3.5156 = -16t^2 + 60t + 3.5156.
Step 5: Rewrite the right side of the equation as a perfect square trinomial by factoring. The right side can be written as -16(t^2 - 3.75t + 3.5156).
Step 6: Rewrite the perfect square trinomial as a squared binomial by taking the square root of the first and last terms of the trinomial and using that to complete the binomial. In this case, we have (t - 1.875)^2.
Step 7: Rewrite the equation with the squared binomial: h(t) + 3.5156 = -16(t - 1.875)^2.
Step 8: Subtract 3.5156 from both sides of the equation to isolate h(t) on the left side. The equation becomes h(t) = -16(t - 1.875)^2 - 3.5156.
Step 9: Now we can see that the maximum height occurs when the squared term (t - 1.875)^2 equals zero. Therefore, to find the maximum height, we need to find the value of t when (t - 1.875)^2 = 0.
Solving for this equation, we find t = 1.875.
Step 10: Substitute the value of t back into the original equation h(t) = 60t - 16t^2 to find the maximum height.
By substituting t = 1.875 into h(t), we get h(1.875) = 60(1.875) - 16(1.875)^2. Calculating this expression, we find h(1.875) = 56.25 feet.
Therefore, the maximum height of the ball after Nick releases it is 56.25 feet.