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Guillermo is a professional deep water free diver. His altitude (in meters relative to sea level), x seconds after diving, is nodeled by g(x)=(1)/(20)x(x-100) How many seconds after diving will Guillermo reach his lowest altitude? seconds

Guillermo is a professional deep water free diver.\newlineHis altitude (in meters relative to sea level), x x seconds after diving, is nodeled by\newlineg(x)=120x(x−100) g(x)=\frac{1}{20} x(x-100) \newlineHow many seconds after diving will Guillermo reach his lowest altitude?\newlineseconds

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Full solution

Q. Guillermo is a professional deep water free diver.\newlineHis altitude (in meters relative to sea level), x x seconds after diving, is nodeled by\newlineg(x)=120x(x−100) g(x)=\frac{1}{20} x(x-100) \newlineHow many seconds after diving will Guillermo reach his lowest altitude?\newlineseconds
  1. Understand the problem: Understand the problem.\newlineWe need to find the time at which Guillermo reaches his lowest altitude after diving. The altitude is given by the function g(x)=120x(x−100)g(x) = \frac{1}{20}x(x - 100).
  2. Determine critical points: Determine the critical points of the function.\newlineTo find the lowest altitude, we need to find the minimum point of the function g(x)g(x). This occurs at the vertex of the parabola represented by the function since the coefficient of x2x^2 is positive, indicating a parabola that opens upwards.
  3. Find vertex x-coordinate: Find the x-coordinate of the vertex.\newlineThe vertex of a parabola in the form of f(x)=ax2+bx+cf(x) = ax^2 + bx + c is given by the formula −b2a-\frac{b}{2a}. However, our function is in the form of g(x)=120x(x−100)g(x) = \frac{1}{20}x(x - 100), which can be rewritten as g(x)=120(x2−100x)g(x) = \frac{1}{20}(x^2 - 100x). Here, a=120a = \frac{1}{20} and b=−10020=−5b = -\frac{100}{20} = -5. So, the x-coordinate of the vertex is −−52⋅(120)-\frac{-5}{2\cdot(\frac{1}{20})}.
  4. Calculate x-coordinate: Calculate the x-coordinate of the vertex.\newlinexx-coordinate = −(−5)/(2∗(1/20))=5/(1/10)=5∗10=50-(-5)/(2*(1/20)) = 5/(1/10) = 5*10 = 50.
  5. Interpret the result: Interpret the result.\newlineThe xx-coordinate of the vertex represents the time in seconds after diving when Guillermo will reach his lowest altitude. Therefore, Guillermo will reach his lowest altitude 5050 seconds after diving.

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