3) To convert the quadratic equation y = 2x² - 24x + 5 to vertex form, we need to complete the square. The vertex form of a quadratic equation is given by y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex.
Let's start by completing the square:
y = 2x² - 24x + 5
= 2(x² - 12x) + 5
To complete the square, we need to take half of the coefficient of x, square it, and add it inside the parentheses. In this case, half of -12 is -6, and (-6)² = 36. So we add 36 inside the parentheses while subtracting 36 outside the parentheses to maintain the equality:
y = 2(x² - 12x + 36 - 36) + 5
= 2((x - 6)² - 36) + 5
= 2(x - 6)² - 72 + 5
= 2(x - 6)² - 67
Now the equation is in vertex form. The vertex of the parabola is given by the coordinates (h, k), so in this case, the vertex is (6, -67).
4) The equation h = -2t² + 9t + 35 represents the height of the toy soldier above the water at time t.
To determine when the toy soldier hits the surface of the canal, we need to find the value of t when h equals zero.
So, we set h = 0 and solve for t:
0 = -2t² + 9t + 35
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = -2, b = 9, and c = 35. Substituting these values into the quadratic formula, we get:
t = (-(9) ± √((9)² - 4(-2)(35))) / (2(-2))
= (-9 ± √(81 + 280)) / (-4)
= (-9 ± √(361)) / (-4)
= (-9 ± 19) / (-4)
So we have two possible solutions for t:
1) t = (-9 + 19) / (-4) = 5/2 = 2.5
2) t = (-9 - 19) / (-4) = 28/4 = 7
Therefore, the toy soldier will hit the surface of the canal at two different times: 2.5 seconds and 7 seconds.