A person pushes a lawn mower
with a force of 42 pounds.
The handle of the lawn mower is 38 degrees
above the horizontal.
So if this is the lawn mower handle,
we know this angle here is 38 degrees.
And we're asked to determine how much downward force
and horizontal force is being exerted.
We can represent the force being applied to the lawn mower
as a vector
where, because the force is 42 pounds,
the magnitude of the vector is 42 pounds.
And we assume the vector is being applied
in the direction of the handle
and therefore the force vector
would look something like this.
Where, again, because the force is 42 pounds,
we know the magnitude of this vector,
we'll call it vector v,
is 42,
which represents the length of the vector.
In order to determine the downward
and horizontal force though,
we need to find the horizontal or x component
as well as the vertical or y component of this vector.
To do this, let's graph this same vector
so the initial point, this point here, is at the origin.
So this vector is equivalent to this vector here.
Where, again, we know the magnitude of the vector is 42.
Now, the direction of the vector is not 38 degrees.
Remember, the direction of a vector
is measured from the positive x-axis.
And therefore, this angle here
is the direction, which we often call angle theta,
which is 180 degrees plus 38 degrees, or 218 degrees.
So the angle theta, the direction of the vector,
is 218 degrees.
In order to determine the downward force
and horizontal force,
we now need to find the x and y components of the vector.
Where this would be the horizontal component of the vector
and this would be the vertical or y component of the vector.
The horizontal component will give us the horizontal force
and the vertical component will give us the downward force.
From here, remember,
if a vector makes an angle theta with the positive x axis,
then the x or horizontal component is equal to
the magnitude of the vector times cosine theta
and the y component or vertical component is equal to
the magnitude of the vector times sine theta.
Which means, for our vector v, the force vector,
the x component is equal to 42 times cosine 218 degrees
and the y component is equal to 42 times
the sine of 218 degrees.
And now we'll go to the calculator
and get decimal approximations for these components.
We will round to two decimal places.
First, let's make sure the calculator is in degree mode.
Press mode and notice how the calculator is in degree mode
because degree is highlighted.
So we go back to the home screen
and we enter 42 cosine 218 degrees, enter.
Rounding to the second decimal place,
because we have a six in the third decimal place we round up
and because we have a nine in the second decimal place,
to the hundredths place value,
this rounds to negative 33.10.
And now for the y component,
we enter 42 sine 218 degrees.
Because we have a seven in the third decimal place,
this rounds to negative 25.86.
The x or horizontal component of negative 33.10
indicates a force in the direction of left horizontally
of approximately 33.10 pounds.
So the horizontal force being exerted
is approximately 33.10 pounds.
We don't include the sign.
And because the y component or vertical component
is approximately negative 25.86,
this indicates a force in the direction of downward
of approximately 25.86 pounds.
So the downward force is approximately 25.86 pounds.
I hope you found this helpful.