By completing the square, determine the point in time when your model predicts Peter’s fortune will turn around. What is the lowest value that Peter’s investment will reach at this time?
my function was v(t)=25x^2-750x+10000
v(9)=25(9)^2-750(9)+10000=5275
the vertex of ax^2+bx+c is at x = -b/2a
In this case, that is 750/50 = 15, not 9.
v(15) = 4375
v(t) = (5x-75)^2 + 4375
so the lowest value possible for v(t) is 4375 at x=15.
It specified "completing the square", so
v(x) = 25x^2 - 750x + 10000 , you had v(t) but no t showed up in your function.
= 25(x^2 - 30x) + 10000
= 25(x^2 - 30x + 225 - 225) + 10000
= 25( (x-15)^2 - 225) + 10000
= 25(x-15)^2 - 5625+10000
= 25(x-15)^2 + 4375
lowest value is 4375 , when x = 15
To determine the point in time when Peter's fortune will turn around, we need to find the vertex of the quadratic function v(t) = 25t^2 - 750t + 10000.
We can complete the square to find the vertex. The vertex form of a quadratic function is v(t) = a(t-h)^2 + k, where (h, k) represents the vertex.
Let's apply the steps to complete the square in order to find the vertex:
1. First, divide the equation by the leading coefficient 25 to make the coefficient of t^2 equal to 1:
v(t) = (25/25)t^2 - (750/25)t + 10000/25
Simplifying, we get:
v(t) = t^2 - 30t + 400
2. Next, group the terms involving t and rewrite the equation:
v(t) = (t^2 - 30t) + 400
3. To complete the square, take half of the coefficient of t (-30/2 = -15) and square it (-15^2 = 225). Add this value inside the parentheses and subtract it outside the parentheses to maintain the equation's balance:
v(t) = (t^2 - 30t + 225) + 400 - 225
4. Now, simplify:
v(t) = (t-15)^2 + 175
The vertex is given by the values (h, k), which in this case is (15, 175).
Therefore, the point in time when Peter's fortune will turn around is at t = 15.
To find the lowest value that Peter's investment will reach at this time, we substitute this value into the function:
v(15) = (15-15)^2 + 175
v(15) = 0^2 + 175
v(15) = 175
Therefore, the lowest value that Peter's investment will reach at this time is 175.