AM frequencies of radio transmissions range from 535-1705 kilohertz. Assume they are an exponential function.

Write an equation for the AM frequencies along a display that is 15 cm long.

To write an equation for AM frequencies along a display that is 15 cm long, we need to determine the rate at which the frequencies change with respect to the display length.

Given that AM frequencies are assumed to follow an exponential function and they range from 535-1705 kilohertz, we can assume that the function is of the form:

f(x) = a * b^(kx)

where f(x) represents the frequency, x represents the display length, a is a constant representing the starting frequency, b is the base of the exponential function, and k is a constant that determines the rate at which the frequencies change.

To find the equation, we need to determine the values of a, b, and k.

Using the information provided, we know that when x is 0 (the starting point), the frequency is 535 kilohertz. Therefore, we have the initial condition:

f(0) = a * b^(k * 0) = 535

Similarly, when x is 15 cm (the endpoint), the frequency is 1705 kilohertz. Therefore, we have another condition:

f(15) = a * b^(k * 15) = 1705

Now, we can solve these two equations to find the values of a, b, and k.

Dividing the second equation by the first equation, we get:

(1705/535) = (a * b^(k * 15))/(a * b^(k * 0))

This simplifies to:

3.19 = b^(k * 15)

Taking the natural logarithm of both sides, we have:

ln(3.19) = ln(b^(k * 15))

Using the logarithmic property, ln(b^x) = x * ln(b), we can simplify this to:

ln(3.19) = (k * 15) * ln(b)

Now, let's denote ln(3.19) as C (a constant):

C = (k * 15) * ln(b)

This equation relates the values of k and b. Note that the values of a and b can be any positive real numbers, as they represent the starting frequency and the base of the exponential function, respectively.

Therefore, the equation for the AM frequencies along a display that is 15 cm long is given by:

f(x) = a * b^(k * x)

where a and b are any positive real numbers, and k is determined by the equation:

C = (k * 15) * ln(b)

To write an equation for the AM frequencies along a display that is 15 cm long, we need to determine the relationship between the frequency of the AM transmissions and their position on the display.

Assuming the AM frequencies follow an exponential function, we can use the exponential growth or decay equation:

y = a * (b^x)

Where:
- y represents the frequency of the AM transmission
- x represents the position on the display
- a is the initial frequency at x = 0
- b is the growth/decay factor

To determine the values of a and b, we need to use the given frequency range.

Given:
- Minimum frequency: 535 kilohertz
- Maximum frequency: 1705 kilohertz

The range of AM frequencies from 535 to 1705 kilohertz spans 1170 kilohertz (1705 - 535).

Since the display is 15 cm long, we can assume a linear relationship between the position on the display (x) and the range of frequencies (1170 kilohertz).

So, the growth/decay factor (b) can be calculated as:

b = (1170 kilohertz) / (15 cm)

Now, we need to convert the units. Since 1 cm equals 10 millimeters (mm) and 1 kilohertz equals 1000 hertz (Hz), we can write:

b = (1170 * 1000 Hz) / (15 * 10 mm)

Simplifying, we get:

b = 78,000 Hz/mm

Now that we have the value of b, we can use it to find the initial frequency (a). To do this, we can use one of the given frequencies as a reference point. Let's use the minimum frequency (535 kilohertz) when x = 0.

Therefore:

y = a * (b^x)
535 kilohertz = a * (78,000 Hz/mm)^0
535 kilohertz = a * 1

Since anything to the power of 0 is equal to 1, we have:

535 kilohertz = a

Now we have both the values of a and b. Therefore, the equation for the AM frequencies along a 15 cm display is:

y = 535 kilohertz * (78,000 Hz/mm)^x