Respuesta :
Answer:
The trip consists of two parts. The rowing part is the hypotenuse of right angled triangle
whose sides are the distance from P to the island, which is 5, and the distance between P and
the landing point of the rowboat on the shore, which is x
so this part of trip is sqrt(25+ x^2)
The 2nd part is the walking part, which is (8-x)
Distance = rate times time (D = rt), so to get the time you have t = D/r. We must divide each
of the trip by the appropriate rate to get the time.
a) T(x) = sqrt(25+x^2)/3 + (8-x)/4
To find minimum time required take derivative of the T(x) function and find it's zeros
T'(x) = x/(3(sqrt(25+x^2)) - 1/4 = 0
x/(3(sqrt(25+x^2)) = 1/4
4x = 3sqrt(25+x^2
16x^2 = 9(25+x^2) = 225 + 9x^2
7x^2 = 225
x^2 = 225/7
x = sqrt(225/7) = 5.669467 miles
T(x) = 3.602386382 hours
Step-by-step explanation:
The point where the boat should be landed can be found by expressing
the distance travelled on the boat and walking as a function of time.
The point where the boat should be landed is the point 3.4 miles from the
point P towards the town.
Reasons:
x represent the distance from point P to the boat landing point.
Therefore, distance of rowing the boat = √((12 - x)² + 3²)
The total time, t, is therefore;
[tex]t = \dfrac{12-x}{4} +\dfrac{\sqrt{x^2 + 3^2} }{3}[/tex]
When the time is minimum, we get;
[tex]\dfrac{dt}{dx} = \dfrac{d}{dx} \left( \dfrac{12-x}{4} +\dfrac{\sqrt{x^2 + 3^2} }{3} \right) = \dfrac{12\cdot \left(-3 + 4 \cdot\dfrac{2 \cdot x }{2\cdot \sqrt{x^2 + 9} } \right)}{144}[/tex]
[tex]\dfrac{12\cdot \left(-3 + 4 \cdot\dfrac{2 \cdot x }{2\cdot \sqrt{x^2 + 9} } \right)}{144} = -\dfrac{1}{4} +\dfrac{x }{3 \cdot \sqrt{x^2 +9} }[/tex]
[tex]\dfrac{x }{3 \cdot \sqrt{x^2 +9} } = \dfrac{1}{4}[/tex]
4·x = 3·√(x² + 9)
16·x² = 9·(x² + 9)
7·x² = 81
[tex]x = \dfrac{9 \cdot \sqrt{7} }{7}[/tex]
x ≈ 3.4 miles.
The point where the boat should be landed is the point approximately 3.4
miles from the point P in the direction of the town.
Learn more here:
https://brainly.com/question/13747147
