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Rui is a professional deep water free diver. His altitude (in meters relative to sea level), x seconds after diving, is modeled by: d(x)=(1)/(2)x^(2)-10 x What is the lowest altitude Rui will reach? meters relative to sea level

Rui is a professional deep water free diver.\newlineHis altitude (in meters relative to sea level), x x seconds after diving, is modeled by:\newlined(x)=12x2−10x d(x)=\frac{1}{2} x^{2}-10 x \newlineWhat is the lowest altitude Rui will reach?\newline □\square meters relative to sea level

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Q. Rui is a professional deep water free diver.\newlineHis altitude (in meters relative to sea level), x x seconds after diving, is modeled by:\newlined(x)=12x2−10x d(x)=\frac{1}{2} x^{2}-10 x \newlineWhat is the lowest altitude Rui will reach?\newline □\square meters relative to sea level
  1. Identify Quadratic Equation: Identify the quadratic equation that models Rui's altitude.\newlineThe given equation is d(x)=(12)x2−10xd(x) = (\frac{1}{2})x^2 - 10x.\newlineHere, a=12a = \frac{1}{2}, b=−10b = -10, and c=0c = 0, since there is no constant term.
  2. Find Vertex Time: Find the xx-coordinate of the vertex of the parabola, which will give us the time at which Rui reaches his lowest altitude.\newlineThe xx-coordinate of the vertex of a parabola given by ax2+bx+cax^2 + bx + c is found using the formula x=−b2ax = -\frac{b}{2a}.\newlineSubstitute a=12a = \frac{1}{2} and b=−10b = -10 into the formula.\newlinex=−−102∗(12)x = -\frac{-10}{2*(\frac{1}{2})}\newlinex=101x = \frac{10}{1}\newlinex=10x = 10
  3. Calculate Lowest Altitude: Calculate the lowest altitude Rui will reach by substituting the xx-coordinate of the vertex into the original equation.d(x)=(12)x2−10xd(x) = \left(\frac{1}{2}\right)x^2 - 10xd(10)=(12)(10)2−10(10)d(10) = \left(\frac{1}{2}\right)(10)^2 - 10(10)d(10)=(12)(100)−100d(10) = \left(\frac{1}{2}\right)(100) - 100d(10)=50−100d(10) = 50 - 100d(10)=−50d(10) = -50

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