A spot of light is made to travel across a computer screen in a straight line so that, a t seconds after starting, it's distance from the left hand edge (d cm) is given by the function d(t) =7t - t² +2. Find the furthest distance the spot light travels and how long it takes to travel this distance.
so you just want the max of d(t)
find the derivative, set it equal to zero, and solve that for t
Once you have the value of t which gives you the maximum, plug it back into d(t) to actually find that distance.
Very straight-forward.
To find the furthest distance the spot light travels, we need to maximize the function d(t).
To find the maximum value of a quadratic function, we can use the vertex formula.
The formula for the vertex of a quadratic function in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants, is given by:
x = -b / (2a)
In our case, the quadratic function is d(t) = 7t - t^2 + 2. Comparing this to the general form, we can see that a = -1, b = 7, and c = 2.
Using the vertex formula, we can find the value of t at which the maximum distance occurs:
t = -b / (2a)
= -7 / (2 * -1)
= 7 / 2
= 3.5
Now, substitute this value of t back into the function d(t) to find the maximum distance:
d(t) = 7t - t^2 + 2
= 7(3.5) - (3.5)^2 + 2
= 24.5 - 12.25 + 2
= 14.25
Therefore, the furthest distance the spot light travels is 14.25 cm.
To find how long it takes to travel this distance, we already know that t = 3.5 seconds.
So, it takes 3.5 seconds for the spot light to travel the furthest distance of 14.25 cm.
To find the furthest distance the spot of light travels, we need to determine the vertex of the quadratic function d(t) = 7t - t² + 2. The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)), where f(x) is the function.
In our case, a = -1, b = 7, and c = 2. Plugging these values into the formula, we have t = -b/2a = -7/(-2) = 7/2.
To find the corresponding distance, we substitute this value into the function d(t):
d(7/2) = 7(7/2) - (7/2)^2 + 2
= 49/2 - 49/4 + 2
= 98/4 - 49/4 + 8/4
= 57/4
Therefore, the furthest distance the spot of light travels is 57/4 cm.
To determine how long it takes to travel this distance, we substitute the time value into the function:
d(t) = 57/4
7t - t^2 + 2 = 57/4
Rearranging the equation, we get a quadratic equation:
4(7t - t^2 + 2) = 57
28t - 4t^2 + 8 = 57
-4t^2 + 28t + 8 - 57 = 0
-4t^2 + 28t - 49 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values a = -4, b = 28, c = -49, we have:
t = (-28 ± √(28^2 - 4(-4)(-49))) / (2(-4))
t = (-28 ± √(784 - 784)) / (-8)
t = (-28 ± √0) / (-8)
t = -28 / -8
t = 7/2
Since time cannot be negative, the spot light reaches the furthest distance after 7/2 seconds, or 3.5 seconds.
Therefore, the furthest distance the spot light travels is 57/4 cm, and it takes 3.5 seconds to travel this distance.