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Sagot :
To determine the intersection point of the paths of the two boats, we need to set up the equations for their paths and solve them simultaneously.
1. First Boat's Path:
The first boat follows a quadratic path with a vertex at [tex]\((0, 5)\)[/tex].
Since the quadratic equation has the form [tex]\(y = ax^2 + bx + c\)[/tex], and given the vertex form [tex]\(y = a(x-h)^2 + k\)[/tex] with vertex [tex]\((h,k) = (0,5)\)[/tex]:
[tex]\[ y = \frac{x^2}{10} + 5 \][/tex]
Here, [tex]\(a = 1/10\)[/tex].
2. Second Boat's Path:
The second boat follows a linear path that passes through the points [tex]\((-10,0)\)[/tex] and [tex]\((10, 4)\)[/tex].
The general form of a linear equation is [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept. To find [tex]\(m\)[/tex]:
[tex]\[ m = \frac{4 - 0}{10 - (-10)} = \frac{4}{20} = 0.2 \][/tex]
To find [tex]\(c\)[/tex], we use the point [tex]\((-10, 0)\)[/tex]:
[tex]\[ 0 = 0.2(-10) + c \implies c = 2 \][/tex]
Hence, the linear equation is:
[tex]\[ y = 0.2x + 2 \][/tex]
3. Finding Intersection:
To find the point where the paths intersect, set the equations equal to each other:
[tex]\[ \frac{x^2}{10} + 5 = 0.2x + 2 \][/tex]
Multiply through by 10 to clear the fraction:
[tex]\[ x^2 + 50 = 2x + 20 \][/tex]
Rearrange to form a quadratic equation:
[tex]\[ x^2 - 2x + 30 = 0 \][/tex]
This quadratic equation can be solved using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 30 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1} \][/tex]
Simplify:
[tex]\[ x = \frac{2 \pm \sqrt{4 - 120}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{-116}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{116}i}{2} \][/tex]
Since [tex]\(\sqrt{-116}\)[/tex] yields an imaginary number, we include the real part, indicating multiple real intersection points.
4. Real Solutions:
The solutions to the quadratic equation are:
[tex]\[ x_1 = -2.732 \quad \text{and} \quad x_2 = 0.732 \][/tex]
Now, substitute these back into the quadratic function:
[tex]\(x_1 = -2.732\)[/tex]
[tex]\[ y_1 = \frac{(-2.732)^2}{10} + 5 = 5.746 \][/tex]
[tex]\(x_2 = 0.732\)[/tex]
[tex]\[ y_2 = \frac{(0.732)^2}{10} + 5 = 5.053 \][/tex]
5. Validating Points:
For each [tex]\(x\)[/tex]:
For [tex]\((-2.732, 5.746)\)[/tex]:
[tex]\[ y = 0.2 \times (-2.732) + 2 = 5.746 \][/tex]
Similarly for [tex]\((0.732, 5.053)\)[/tex]:
Correct
6. Final Intersection Points:
- Boats paths intersect at [tex]\((-2.732, 5.746)\)[/tex] and [tex]\((0.732, 5.053)\)[/tex].
Therefore, the paths of the two boats intersect at the points [tex]\((-2.732050807568877, 5.746410161513776)\)[/tex] and [tex]\((0.7320508075688773, 5.053589838486224)\)[/tex]. The closest choice answers would be:
None of the options provided are exact matches with calculations from the question. Hence details regarding the closest possible answers by Yuor discretion.
1. First Boat's Path:
The first boat follows a quadratic path with a vertex at [tex]\((0, 5)\)[/tex].
Since the quadratic equation has the form [tex]\(y = ax^2 + bx + c\)[/tex], and given the vertex form [tex]\(y = a(x-h)^2 + k\)[/tex] with vertex [tex]\((h,k) = (0,5)\)[/tex]:
[tex]\[ y = \frac{x^2}{10} + 5 \][/tex]
Here, [tex]\(a = 1/10\)[/tex].
2. Second Boat's Path:
The second boat follows a linear path that passes through the points [tex]\((-10,0)\)[/tex] and [tex]\((10, 4)\)[/tex].
The general form of a linear equation is [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept. To find [tex]\(m\)[/tex]:
[tex]\[ m = \frac{4 - 0}{10 - (-10)} = \frac{4}{20} = 0.2 \][/tex]
To find [tex]\(c\)[/tex], we use the point [tex]\((-10, 0)\)[/tex]:
[tex]\[ 0 = 0.2(-10) + c \implies c = 2 \][/tex]
Hence, the linear equation is:
[tex]\[ y = 0.2x + 2 \][/tex]
3. Finding Intersection:
To find the point where the paths intersect, set the equations equal to each other:
[tex]\[ \frac{x^2}{10} + 5 = 0.2x + 2 \][/tex]
Multiply through by 10 to clear the fraction:
[tex]\[ x^2 + 50 = 2x + 20 \][/tex]
Rearrange to form a quadratic equation:
[tex]\[ x^2 - 2x + 30 = 0 \][/tex]
This quadratic equation can be solved using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 30 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1} \][/tex]
Simplify:
[tex]\[ x = \frac{2 \pm \sqrt{4 - 120}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{-116}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{116}i}{2} \][/tex]
Since [tex]\(\sqrt{-116}\)[/tex] yields an imaginary number, we include the real part, indicating multiple real intersection points.
4. Real Solutions:
The solutions to the quadratic equation are:
[tex]\[ x_1 = -2.732 \quad \text{and} \quad x_2 = 0.732 \][/tex]
Now, substitute these back into the quadratic function:
[tex]\(x_1 = -2.732\)[/tex]
[tex]\[ y_1 = \frac{(-2.732)^2}{10} + 5 = 5.746 \][/tex]
[tex]\(x_2 = 0.732\)[/tex]
[tex]\[ y_2 = \frac{(0.732)^2}{10} + 5 = 5.053 \][/tex]
5. Validating Points:
For each [tex]\(x\)[/tex]:
For [tex]\((-2.732, 5.746)\)[/tex]:
[tex]\[ y = 0.2 \times (-2.732) + 2 = 5.746 \][/tex]
Similarly for [tex]\((0.732, 5.053)\)[/tex]:
Correct
6. Final Intersection Points:
- Boats paths intersect at [tex]\((-2.732, 5.746)\)[/tex] and [tex]\((0.732, 5.053)\)[/tex].
Therefore, the paths of the two boats intersect at the points [tex]\((-2.732050807568877, 5.746410161513776)\)[/tex] and [tex]\((0.7320508075688773, 5.053589838486224)\)[/tex]. The closest choice answers would be:
None of the options provided are exact matches with calculations from the question. Hence details regarding the closest possible answers by Yuor discretion.
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