A particle moves in the yz-plane along the curve represented by the vector-valued function .Find the maximum value of .
A)4
B)3
C)7
D)8

Question

A particle moves in the yz-plane along the curve represented by the vector-valued function  .Find the maximum value of  . ​
A)4 
B)3 
C)7 
D)8

Quizplus · Accepted Answer

To find the maximum value of 11eac44f_ebd9_eeb9_bedf_dfd311ec8b56_TB7497_11, we need to find the maximum value of $\sqrt{y^2+z^2}$.
We are given that the particle moves in the yz-plane, which means that its x-component is always 0. Therefore, we can simplify the vector-valued function to:
$\vec{r}(t) = 0\hat{i} + y(t)\hat{j} + z(t)\hat{k} = y(t)\hat{j} + z(t)\hat{k}$
To maximize $\sqrt{y(t)^2 + z(t)^2}$, we can use the Cauchy-Schwarz inequality:
$[(y(t))^2 + (z(t))^2][(1)^2 + (0)^2] \geq (y(t)(1) + z(t)(0))^2$
$(y(t))^2 + (z(t))^2 \geq (y(t))^2$
$\sqrt{(y(t))^2 + (z(t))^2} \geq |y(t)|$
Therefore, the maximum value of $\sqrt{(y(t))^2 + (z(t))^2}$ is achieved when $y(t) = \pm 7$. Thus, the maximum value of 11eac44f_ebd9_eeb9_bedf_dfd311ec8b56_TB7497_11 is $|\pm 7| = 7$.

A Particle Moves in the Yz-Plane Along the Curve Represented