At the winter solstice, the power, in watts, received from the Sun on each square metre of Earth's surface can be modelled by the formula
P = 1000sin (x + 113.5 degrees),
where x represents the angle of latitude in the northern hemisphere.
a) Determine the angle of latitude at which the power level drops to 0.
b) Determine the angle of latitude at which the power level is a maximum. Explain what is meant by the negative sign.
a) To find the angle of latitude at which the power level drops to 0, we need to solve the equation for P = 0:
0 = 1000 * sin(x + 113.5°)
Now, to solve for x, we can use the inverse of the sine function, which is arcsin:
arcsin(0) = arcsin(sin(x + 113.5°))
Since sin(x) = 0 when x = 0, we have:
x + 113.5° = 0
Therefore, the angle of latitude at which the power level drops to 0 is -113.5°.
b) To find the angle of latitude at which the power level is a maximum, we need to find the maximum value of the sine function. The maximum value of sine is 1, and it occurs when the angle is 90° (or π/2 radians).
In the given formula, the angle x represents the angle of latitude in the northern hemisphere. Since the northern hemisphere ranges from 0° to 90° latitude, the maximum power level occurs at 90° latitude.
Now, let's address the negative sign in the formula. The negative sign represents a phase shift or a shift in the original sine function. In this case, it means that the power level is at its maximum when the angle of latitude is 90° in the northern hemisphere.
a) To determine the angle of latitude at which the power level drops to 0, we need to set the power equation equal to 0 and solve for x.
P = 1000sin(x + 113.5 degrees)
0 = 1000sin(x + 113.5 degrees)
To solve for x, we need to isolate the sin(x + 113.5 degrees) term.
sin(x + 113.5 degrees) = 0
To find the angle whose sine is 0, we know that sin 0 = 0 and sin 180 = 0. Therefore, we need to find the values of x that satisfy these conditions.
x + 113.5 degrees = 0 or x + 113.5 degrees = 180 degrees
Solving for x in each equation:
x = -113.5 degrees or x = 66.5 degrees
So, the angle of latitude at which the power level drops to 0 is either -113.5 degrees or 66.5 degrees.
b) To determine the angle of latitude at which the power level is a maximum, we need to analyze the equation and consider the properties of the sine function.
In the given equation, we have P = 1000sin(x + 113.5 degrees).
The maximum value for the sine function is 1, and it occurs when the angle is either 90 degrees or -90 degrees.
In our equation, we see that the angle of latitude x is being added to 113.5 degrees. So, if we subtract 113.5 degrees from 90 degrees and -90 degrees, we can find the angle of latitude that gives the maximum power level.
90 degrees - 113.5 degrees = -23.5 degrees
-90 degrees - 113.5 degrees = -203.5 degrees
However, since we are considering angles in the northern hemisphere, where latitudes are positive, we ignore the negative angle.
So, the angle of latitude at which the power level is a maximum is -23.5 degrees. The negative sign indicates that this latitude is below the equator.
Set power to zero.
sin(x+113.5) so x must be 180-113.5
Maximum? x must be 90-113.5
I am quite uncertain what you are missing here.