When a softball player swings a bat, the amount of energy E(t), in joules, that is transferred to the bat can be approximated by the function
E(t) = −279.67t2 + 82.86t
where 0 ¡Ü t ¡Ü 0.3 and t is measured in seconds. According to this model, what is the maximum energy of the bat? Round to the nearest tenth of a joule.
_______ joules
E(t) is just a parabola. I'm sure you know how to find the vertex, where E is a maximum.
To find the maximum energy of the bat, we need to determine the vertex of the quadratic function E(t) = -279.67t^2 + 82.86t.
The formula for finding the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by:
x = -b / (2a)
In our case, a = -279.67 and b = 82.86. Plugging these values into the formula, we get:
t = -82.86 / (2 * -279.67)
Simplifying further:
t = -82.86 / (-559.34)
t ≈ 0.148
Now, substitute this value of t into the function E(t) to find the maximum energy:
E(t) = -279.67(0.148)^2 + 82.86(0.148)
Calculating this expression:
E(t) ≈ -279.67(0.0219) + 12.247
E(t) ≈ -6.104 + 12.247
E(t) ≈ 6.143
Therefore, the maximum energy of the bat, according to this model, is approximately 6.1 joules.
To find the maximum energy of the bat, we need to determine the highest point on the graph of the function E(t).
The function E(t) is a quadratic equation in the form of y = ax^2 + bx + c, where a = -279.67, b = 82.86, and c = 0.
To find the maximum point on the graph, we can use the formula for the x-coordinate of the vertex of a quadratic function, which is given by x = -b/2a.
In this case, x = -(82.86) / (2*(-279.67)). Calculating this expression, we get x = 0.148 seconds.
Now, let's substitute this value of t back into the equation E(t) to find the maximum energy.
E(t) = -279.67t^2 + 82.86t
E(0.148) = -279.67 * (0.148)^2 + 82.86 * 0.148
E(0.148) = -61.8 + 12.2
The maximum energy of the bat is thus -61.8 + 12.2 = -49.6 joules.
Since energy cannot be negative, we discard the negative sign and take the absolute value. Therefore, the maximum energy of the bat is 49.6 joules (rounded to the nearest tenth).