Question

a remote control airplane descends at a rate of two feet per second.After 3 seconds it is 67 feet above the ground. Write the equation in point slope form that models the situation. Then find the height of the plane after eight seconds

Answer 1

• The equation in point slope form is:

`h-67=-2(x-3)`

where h is the height of the plane above ground and t is the time.

• The height of the plane after eight seconds is: 57 feet

Explanation:

Let h denote the height of the plane above the ground.

and t denote the time.

Now, it is given that:

a remote control airplane descends at a rate of two feet per second.

This means that the rate is : -2

Also,

After 3 seconds it is 67 feet above the ground.

This means that the equation which models this situation passes through (3,67).

Hence, by using the point-slope form of the line.

i.e. any line passing through (a,b) and having slope m is given by:

`y-b=m(x-a)`

Here we have:

y=h x=t , m=-2 and (a,b)=(3,67).

Hence, we have the equation as follows:

`h-67=-2(x-3)`

Now, we are asked to find the height of the plane after t seconds i.e. we are asked to find the value of h when t=8 seconds.

`h-67=-2(8-3)\\\\i.e.\\\\h-67=-2(5)\\\\i.e.\\\\h-67=-10\\\\i.e.\\\\h=-10+67\\\\i.e.\\\\h=57\ \text{feet}`

Answer 2

The equation modeling this problem is given by:
y = mx + b
Where,
m: rate of change
x: time in seconds
b: initial height.
We must find the constants m and b.
For m we have:
"descends at a rate of two feet per second":
m = -2
For b we have:
"After 3 seconds it is 67 feet above the ground":
y = -2x + b
67 = -2 (3) + b
67 = -6 + b
67 + 6 = b
b = 73
The equation of the line is:
y = -2x + 73
For eight seconds we evaluate x = 8 in the function:
y = -2 (8) +73
y = -16 + 73
y = 57 feet
Answer:
The equation in point slope form that models the situation is:
y = -2x + 73
The height of the plane after eight seconds is:
y = 57 feet