Answer:
Question 1:
![\vec V=-32.6mph\cdot \hat i+33.8mph\cdot \hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/vk2h19e4uac8cpzymqjyuri6zhr6zthpvi.png)
Question 2:
![\vec V_b=52.6mph\hat i-33.0mph\hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/mktfyg32w7mq202dka644w7m6foqpagtiq.png)
Step-by-step explanation:
1. Balloon moving at 47 mph on a bearing of 134º
The x-component of a vector is the magnitude of the vector multiplied by the cosine of the angle.
The y-component of a vector is the magnitufe of the vector multiplied by the sine of the angle.
![\vec V=47mph\cdot cos(134\º)\hat i+47mph\cdot sin(134\º)\hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/6hji3south40raajp5vv724o6u2yv2zxz9.png)
![\vec V=-32.6mph\cdot \hat i+33.8mph\cdot \hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/vk2h19e4uac8cpzymqjyuri6zhr6zthpvi.png)
2. Balloon moving at 51mph on a bearing of 341º, with a wind blowing at 17mph on a bearing of 285º
First find, the component forms of the initial velocity of the balloon and of the wind. Then, add the two velocities.
i) Initial velocity of the balloon: Vb,i
![\vec V_(b,i)=51mph\cdot cos(341\º) \hat i+51mph\cdot sin(341\º)\hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/h17s5vecy0o9b7r9kxjyyakeqv9axoc1ki.png)
![\vec V_(b,i)=48.2mph\cdot \hat i-16.6mph\cdot \hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/sgvbjf7utrgel6k8sgmy73ebs5elpeq3jj.png)
ii) Wind velocity: Vw
![\vec V_w=17mph\cdot cos(285\º)\hat i+17mph\cdot sin(285\º)\hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/q6d89slc3q8gkd4zoirlb9jfwmsnzu5c3c.png)
![\vec V_w=4.4mph\cdot \hat i-16.4mph\cdot \hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/g6etbvg5w7jgnvtmqfgzxyt8gjobptcvl4.png)
iii) Add the two velocities
Add the corresponding components:
![\vec V_b=(48.2mph+4.4mph)\cdot \hat i+(-16.6mph-16.4mph)\cdot \hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/wd5clwj6us14u3lcrnai085mq8pibohcwm.png)
![\vec V_b=52.6mph\hat i-33.0mph\hat j](https://img.qammunity.org/2021/formulas/mathematics/high-school/mktfyg32w7mq202dka644w7m6foqpagtiq.png)