An object is launched at 19.6 meters per second from a 58.8 meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = -4.9t2+ 19.6t + 58.8, where s is in meters.
a. When does the object strike the ground?
b. What was its maximum height?
c. How long will it take to reach its maximum height?
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a. 6 seconds
To solve the given questions, let's go step by step:
a. When does the object strike the ground?
To determine when the object will strike the ground, we need to find the value of t when the height is zero (since the ground level corresponds to zero height).
The equation for the object's height is s(t) = -4.9t^2 + 19.6t + 58.8.
Setting s(t) = 0 and solving for t:
0 = -4.9t^2 + 19.6t + 58.8
This is a quadratic equation. We can use either factoring, completing the square, or the quadratic formula to solve for t. In this case, let's use the quadratic formula.
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation:
t = (-(19.6) ± √((19.6)^2 - 4(-4.9)(58.8))) / (2(-4.9))
Simplifying the equation will give us two possible values for t, one positive and one negative. However, since we are dealing with time, a negative value is not relevant in this case. Therefore, we only consider the positive value of t.
Calculating t using the quadratic formula will give us the time when the object strikes the ground.
b. What was its maximum height?
To find the maximum height of the object, we need to identify the vertex of the parabolic curve described by the equation.
The equation for the object's height is s(t) = -4.9t^2 + 19.6t + 58.8.
The vertex of a quadratic equation in the form f(x) = ax^2 + bx + c is given by the coordinates (h, k), where h and k can be found using the formulas:
h = -b / (2a), and k = f(h).
In our case, a = -4.9, b = 19.6, and c = 58.8.
Substituting these values into the formulas, we can find the maximum height.
c. How long will it take to reach its maximum height?
The time it takes for the object to reach its maximum height is the time taken when it reaches the vertex of the parabolic curve described by the equation.
Using the vertex formula, we can find the time at which the object reaches its maximum height.
Remember, for both b. and c., the maximum height and the time to reach it can be calculated using the h value found in question b.
Let's calculate the answers for a, b, and c based on the given equation and the explanations provided above.