Question

un professionnel en médecine sportive analyse le niveau de fatigue musculaire dune athlète lors dune activité physique de 40 min. lors des 60 premières minutes, le niveau de fatigue musculaire, f(x), est donné par la règle f(x)= -1,4|x - 40|+ 80, où x représente le temps écoulé (en min) depuis le début de lactivité physique. ensuite, il peut être modélisé par une fonction racine carrée ayant son sommet au point dintersection des deux fonctions et passant par le point (85, 32). combien de temps après la fin de lactivité lathlète retrouvera-t-elle son niveau de fatigue musculaire initial? -1,4|x - 40|+ 80 = 0 -1,4|10 - 40|+ 80 = 24 y = a√(x - h)+ k 85 = 0,4√(32 - 60)+ k a=(y2 - y1)/(x2 - x1)=(30 - 32)/(90 - 85)= -2/5 = -0,4

Explanation:

Step1: Find the initial fatigue level

When \(x = 0\), substitute into \(f(x)=- 1.4|x - 40|+80\).
\[ \begin{align*} f(0)&=-1.4|0 - 40|+80\\ &=-1.4\times40 + 80\\ &=-56+80\\ &=24 \end{align*} \]

Step2: Determine the square - root function

The square - root function is of the form \(y = a\sqrt{x - h}+k\). The vertex (sommet) is at the intersection of the two functions. First, find the value of \(f(x)\) at \(x = 60\) for \(y=-1.4|x - 40|+80\).
\[ \begin{align*} f(60)&=-1.4|60 - 40|+80\\ &=-1.4\times20+80\\ &=-28 + 80\\ &=52 \end{align*} \]
The vertex of the square - root function is at the intersection of the two functions. Let the square - root function \(y=a\sqrt{x - h}+k\). We know the vertex \((h,k)\) is at the intersection of the two functions. After some analysis (using the point \((85,32)\) on the square - root function), assume the vertex of the square - root function is at the point where the first function changes its form. Let's find the square - root function. We know that the square - root function \(y=a\sqrt{x - h}+k\) passes through \((85,32)\).
We set up the equation \(32=a\sqrt{85 - h}+k\). Since the vertex is at the intersection of the two functions, we first find the intersection point of the two - part function.
The first function \(y=-1.4|x - 40|+80\).
We want to find when the square - root function \(y=a\sqrt{x - h}+k\) reaches \(y = 24\) (the initial fatigue level).
Let the square - root function be \(y=a\sqrt{x - h}+k\). We know it passes through \((85,32)\) and assume the vertex is at the intersection of the two functions.
We first find the equation of the square - root function. The general form of a square - root function with vertex \((h,k)\) is \(y=a\sqrt{x - h}+k\).
We know that when \(x = 85,y = 32\).
Let's assume the vertex \((h,k)\) is at the intersection of the two functions. We find that the first function \(y=-1.4|x - 40|+80\) and the second - part square - root function.
We know that the square - root function passes through \((85,32)\) and we want \(y = 24\).
\[ \begin{align*} 24&=a\sqrt{x - h}+k\\ 32&=a\sqrt{85 - h}+k \end{align*} \]
Subtracting the first equation from the second gives \(8=a(\sqrt{85 - h}-\sqrt{x - h})\).
We know that for the first function \(f(x)=-1.4|x - 40|+80\), when \(x = 0,f(0)=24\).
For the square - root function \(y=a\sqrt{x - h}+k\), substituting \((85,32)\) gives \(32=a\sqrt{85 - h}+k\).
We assume the vertex of the square - root function is at the intersection of the two functions. Let's find the intersection of the first function \(y=-1.4|x - 40|+80\) and \(y = 24\).
\[ \begin{align*} -1.4|x - 40|+80&=24\\ -1.4|x - 40|&=- 56\\ |x - 40|& = 40 \end{align*} \]
We get \(x=0\) or \(x = 80\).
For the square - root function \(y=a\sqrt{x - h}+k\) passing through \((85,32)\), assume the vertex \((h,k)\) is at the intersection of the two functions. Let \(y=a\sqrt{x - h}+k\), substituting \((85,32)\) gives \(32=a\sqrt{85 - h}+k\).
We want \(y = 24\), so \(24=a\sqrt{x - h}+k\).
Subtracting gives \(8=a(\sqrt{85 - h}-\sqrt{x - h})\).
We know that the square - root function can be written as \(y=a\sqrt{x - 80}+32\) (assuming the vertex is at the intersection of the two functions).
Substituting \(y = 24\) into \(y=a\sqrt{x - 80}+32\) gives:
\[ \begin{align*} 24&=a\sqrt{x - 80}+32\\ a\sqrt{x - 80}&=-8 \end{align*} \]
Since the function passes through \((85,32)\), we first find \(a\).
\[ \begin{align*} 32&=a\sqrt{85 - 80}+32\\ 32&=a\sqrt{5}+32\\ a&=0 \end{align*} \]
This is wrong. Let's start over.
The first function \(f(x)=-1.4|x - 40|+80\) for \(0\leq x\leq60\).
The square - root function \(y=a\sqrt{x - h}+k\) with vertex at the intersection of the two functions and passing through \((85,32)\).
The intersection of \(y=-1.4|x - 40|+80\) and \(y=a\sqrt{x - h}+k\) gives us the vertex of the square - root function.
First, for \(y=-1.4|x - 40|+80\), when \(x = 60\), \(y=-1.4\times20 + 80=52\).
The square - root function \(y=a\sqrt{x - h}+k\) passes through \((85,32)\). Let the vertex be \((h,k)\).
We know \(32=a\sqrt{85 - h}+k\).
We want to find when \(y = 24\).
The first function \(f(x)=-1.4|x - 40|+80\), when \(x = 0,f(0)=24\).
For the square - root function \(y=a\sqrt{x - h}+k\), substituting \((85,32)\) gives \(32=a\sqrt{85 - h}+k\).
We know that the square - root function is of the form \(y=a\sqrt{x - 60}+52\) (since the change occurs at \(x = 60\)).
Substituting \((85,32)\) into \(y=a\sqrt{x - 60}+52\):
\[ \begin{align*} 32&=a\sqrt{85 - 60}+52\\ 32&=a\sqrt{25}+52\\ 32&=5a+52\\ 5a&=-20\\ a&=- 4 \end{align*} \]
So the square - root function is \(y=-4\sqrt{x - 60}+52\).
We want to find \(x\) when \(y = 24\).
\[ \begin{align*} 24&=-4\sqrt{x - 60}+52\\ -4\sqrt{x - 60}&=24 - 52\\ -4\sqrt{x - 60}&=-28\\ \sqrt{x - 60}&=7\\ x-60&=49\\ x&=109 \end{align*} \]
The physical activity lasts \(40\) minutes. The time after the end of the activity is \(109-40 = 69\) minutes.

Answer:

69 minutes