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Chapter 11: Q. 58 (page 873)

Annie is making a crossing from south to north between two islands. The distance between the islands is 2 miles. A current in the channel pushes her off her line and slows her progress towards the opposite island. Her position t hours after leaving the southern island is
$(0.255 t^{3} - 0.479 t^{2}, 2 t - 0.958 t^{2} + 0.511 t^{3})$
Space coordinates are measured in miles from her starting position; time is measured in hours.
(a) What is Annie’s actual velocity vector at any time t?
(b) Where is Annie’s speed a minimum? What is her speed at that point?

Short Answer

Expert verified

$v' (0.626) = <- 0.00022, 0.0033>$

Step by step solution

01

Given Expression

The objective of this problem is to find Annie's actual velocity vector and her minimum speed

Her position after leaving the southern island is

$r (t) = <0.255 t^{3} - 0.479 t^{2}, 2 t - 0.958 t^{2} + 0.51 t^{3}>$

02

Subpart (a) Step 1: Calculation

For velocity differentiate the position vector- valued function

$v (t) = r' (t) = <0.765 t^{2} - 0.958 t, 2 - 1.916 t + 1.533 t^{2}>$

03

Subpart (b) Step 2: Calculation

$v' (t) = r' (t) = <1.53 t - 0.958, - 1.916 + 3.066 t> 0 = <1.53 t - 0.958, - 1.916 + 3.066 t> 1.53 t - 0.958 = 0, - 1.916 + 3.066 t = 0$

Therefore role="math" $t = 0.626, 0.625$

Minimum Speed

$v' (0.626) = <- 0.00022, 0.0033>$

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Most popular questions from this chapter

Let $x_{1} (t)$ , $x_{2} (t)$ , $y_{1} (t)$ , and $y_{2} (t)$ be differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" $r_{1} (t) = <x_{1} (t), y_{1} (t)>$ and role="math" localid="1649579122624" $r_{2} (t) = <x_{2} (t), y_{2} (t)>$ is a differentiable scalar function.

Let y = f(x). State the definition for the continuity of the function f at a point c in the domain of f .

Given a vector-valued function r(t) with domain $ℝ,$ what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

$Explain why we do not need to explicitly use a limit in the definition for the definite integral of a vector function r (t) = <x (t), y (t), z (t)> .$

Let $r (t) = (x (t), y (t)), t \in [a, \infty),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in a circle centered at the origin. (Hint: Graph the functions $r_{1} (t) = (\frac{1}{t}, \frac{1}{t})$ and $r_{2} (t) = (t, t),$ both with domain [1,∞).)

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