On the reality TV show Last Man Standing, a package is thrown from a plane to the ocean below. The contestants must swim to the package to receive the "free pass" located inside the package. The path that the package follows can be modelled by the quadratic function d(t)= -4.9t^2 + 10t + 1200, where d represents the distance, in meters, that the package travels and t is the time in seconds.
A.) How long will it take for the package to reach the water?
B.) Determine the domain and range of the function
If someone could please show me step by step how to do this, that would be awesome. Thanks in advance.
your function is modelled by a parabola which opens downwards.
when the package reaches the water, h(t) = 0
-4.9t^2 + 10t + 1200 = 0
t = (-10 ± √23620)/-9.8
= -14.662 or 16.7029
We will reject the negative time, so
it took appr 16.7 seconds for the package to reach the water
for the domain , since we would only use numbers greater than zero
0 ≤ t ≤ 16.7029
for the range, we need the vertex,
the t of the vertex = -10/-9.8 = 1.020408
plug that into the function:
h(1.020408) = -4.9(1.020408)^2 + 10(1.020408) + 1200
= 1205.1 m
range: 0 ≤ y ≤ 1205.1
To find the time it takes for the package to reach the water (question A), we need to find the value of t when the distance, d(t), equals zero. This is because the package reaches the water at the point where its distance is zero.
So, we can set up the equation:
-4.9t^2 + 10t + 1200 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = -4.9, b = 10, and c = 1200. Now let's plug these values into the quadratic formula:
t = (-10 ± √(10^2 - 4 * -4.9 * 1200)) / (2 * -4.9)
Calculating this expression, we get two solutions: t ≈ -7.15 and t ≈ 24.55.
Notice that time (t) cannot be negative in this context, so we discard the negative solution. Therefore, it will take approximately 24.55 seconds for the package to reach the water.
Moving on to question B, let's determine the domain and range of the function d(t) = -4.9t^2 + 10t + 1200.
The domain of a quadratic function is all real numbers, so this means t can take any value.
To find the range, we look at the coefficient of the quadratic term (t^2). Since it is negative (-4.9), it means the parabola opens downward. This tells us that the range is bounded above. To find the maximum value, we can use the formula for the x-coordinate of the vertex:
t = -b / (2a)
Using our values of a = -4.9 and b = 10, we find:
t = -10 / (2 * -4.9) ≈ 1.02
Now, substitute this value (t = 1.02) back into the equation to find the maximum distance:
d(t) = -4.9 * (1.02)^2 + 10 * 1.02 + 1200 ≈ 1205.16
Therefore, the range of the function is approximately (0, 1205.16]. The function's range is all real numbers greater than or equal to 0, up to the maximum distance reached of approximately 1205.16 meters.
In summary:
A.) It will take approximately 24.55 seconds for the package to reach the water.
B.) The domain of the function is all real numbers, and the range is approximately (0, 1205.16].
A) To find the time it takes for the package to reach the water, we need to determine when the distance is equal to zero (since the package reaches the water when it has no distance left to travel).
Step 1: Set d(t) = 0:
-4.9t^2 + 10t + 1200 = 0
Step 2: Solve the quadratic equation for t. There are a few ways to do this, and one common method is using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -4.9, b = 10, and c = 1200. Substituting these values into the quadratic formula, we get:
t = (-10 ± √(10^2 - 4*(-4.9)*1200)) / (2*(-4.9))
Simplifying further:
t = (-10 ± √(100 + 23520)) / (-9.8)
t = (-10 ± √23620) / (-9.8)
Step 3: Calculate the square root: √23620 ≈ 153.704
t ≈ (-10 ± 153.704) / (-9.8)
Now, we have two potential values for t:
t1 ≈ (-10 + 153.704) / (-9.8) ≈ 14.340 seconds
t2 ≈ (-10 - 153.704) / (-9.8) ≈ 128.115 seconds
Since time cannot be negative in this context, we take the positive value, t1 ≈ 14.340 seconds.
So, it will take approximately 14.340 seconds for the package to reach the water.
B) The domain of the function is the set of all possible input values (t) for the function. In this case, t represents time, so the domain is all real numbers, or (-∞, +∞).
The range of the function is the set of all possible output values (d) for the function. In this case, d represents distance, so the range will depend on the context of the problem. However, based on the given quadratic function, the range can be determined by analyzing the opening direction of the parabolic curve.
Since the coefficient of the quadratic term (-4.9t^2) is negative, the parabola opens downwards. This means that the maximum value of the function occurs at the vertex. In this case, the vertex represents the maximum distance the package will travel before reaching the water.
The x-coordinate of the vertex is given by the formula: x = -b / (2a)
In our case, a = -4.9 and b = 10. Substituting these values:
x = -10 / (2*(-4.9)) ≈ 1.02
So, the vertex occurs at t ≈ 1.02 seconds.
Now, we can substitute this value back into the original function to find the maximum distance:
d(1.02) = -4.9(1.02)^2 + 10(1.02) + 1200 ≈ 1204.579
Therefore, the maximum distance the package will travel before reaching the water is approximately 1204.579 meters.
Based on this information, we can determine the range of the function: (-∞, 1204.579]. This means that the package will travel distances ranging from negative infinity to 1204.579 meters (including the endpoint 1204.579).