Phoebe and Tucker are 48m apart. Phoebe looks up at an angle of 43 degrees to see a hot air balloon hovering in the sky directly between her and Tucker. Tucker looks up at an angle of 31 degrees to see the hot air balloon.
A.) Explain how it could be determined who is closer to the balloon without doing any calculations.
B.) Determine the distance between the balloon and the person nearest the balloon. (I think I have to use sine but I'm not sure how to do this.)
A) To determine who is closer to the balloon without performing any calculations, we can consider the angles at which Phoebe and Tucker are looking up. Since Phoebe's angle is steeper (43 degrees) compared to Tucker's angle (31 degrees), it means that Phoebe is looking up at a higher angle, and therefore she must be closer to the balloon. This assumption is based on the principle that the steeper the angle of elevation, the closer the object is to the observer.
B) To determine the distance between the balloon and the person nearest to it, you are correct in using the sine function. You can utilize the concept of trigonometry and the properties of right-angled triangles.
Let's say the distance between Phoebe and the balloon is represented by d1, and the distance between Tucker and the balloon is represented by d2.
Since Phoebe looks up at an angle of 43 degrees, and Tucker looks up at an angle of 31 degrees, we can define the following relationships:
For Phoebe:
Sine of the angle = Opposite side / Hypotenuse
Sine(43 degrees) = d1 / 48m
For Tucker:
Sine of the angle = Opposite side / Hypotenuse
Sine(31 degrees) = d2 / 48m
To find the distance between the balloon and the person nearest to it, we need to compare d1 and d2, and choose the smaller value.
To calculate d1:
Multiply 48m by the sine of 43 degrees: d1 = 48m * sin(43 degrees)
To calculate d2:
Multiply 48m by the sine of 31 degrees: d2 = 48m * sin(31 degrees)
Now, compare d1 and d2, and the smaller distance represents the person who is closer to the balloon.
A.) Without doing any calculations, we can determine who is closer to the balloon based on their respective angles of elevation. Since Phoebe's angle of elevation (43 degrees) is greater than Tucker's angle of elevation (31 degrees), it means Phoebe is closer to the balloon. This is because the steeper the angle of elevation, the closer the person is to the object being observed.
B.) To determine the distance between the balloon and the person nearest to it, we can use the concept of trigonometry. Specifically, we can use the sine function to find the distance.
Let's denote:
- The distance between Phoebe and the balloon as x.
- The distance between Tucker and the balloon as y.
- The height of the balloon as h.
We are given:
- The angle of elevation for Phoebe, θ1 = 43 degrees.
- The angle of elevation for Tucker, θ2 = 31 degrees.
Using the sine function, we can write the following equations:
For Phoebe:
sin(θ1) = h / x
For Tucker:
sin(θ2) = h / y
Since we want to find the distance between the balloon and the person nearest to it, which is either x or y, we need to rearrange the equations to solve for x and y.
For Phoebe:
x = h / sin(θ1)
For Tucker:
y = h / sin(θ2)
Now, if we can find the height of the balloon, h, we can substitute it into either equation to find the distance between the balloon and the person nearest to it.
Make a sketch, with P for Phoebe and T for Tucker as a horizontal line, so that PT = 45 m
Let B be the position of the balloon, so that angle BPT = 43° and angle BTP = 31°
Let AB be the height of the balloon with A directly below B on PT
sin 43° = AB/BP ---> BP = AB/sin43
sin 31° = AB/BT ---> BT = AB/sin31
now we know, or should know, that sin 43>sin31
So AB divided by the larger number would produce the smaller answer.
So BP < BT
(notice I did not do any calculations)
To actually find PB, let PA = x, the AT = 48-x
tan 43 = AB/x
AB = xtan43
tan31 = AB/(48-x)
AB = (48-x)tan31
so xtan43 = 48-xtan31
xtan43 + tan31 = 48
x(tan43+tan31) = 48
x = 48/(tan43+tan31)
now cos 43 = x/BP
BP = x/cos43
= (48/(tan43 + tan31) )(1/cos43)
only now do we do any calculation ....
I get BP = 42.8 m
check my work please