Introduction

The range of a broadcast signal from a cellphone tower is a critical factor for network coverage. This paper explores how to determine the part of a straight highway where cell phones can effectively use their towers. Specifically, we will use a mathematical approach to solve a real-world problem by intersecting the signal range with the highway geometrically.

Mathematical Background

We begin with a cellphone tower whose broadcast signal is bounded by a circle with the equation (x^2 y^2 1850). Simultaneously, the highway is modeled by the linear equation (y -frac{1}{4}x 40). Our task is to find the points where the highway intersects the circle and the lengths of the highway which fall within the coverage range of the cellphone tower.

Mathematical Approach

Step 1: Substitution

We need to substitute the equation of the highway into the circle equation. The equation of the highway is given by:

[ y -frac{1}{4}x 40 ]

Substituting this into the circle equation gives:

[ x^2 left(-frac{1}{4}x 40right)^2 1850 ]

Step 2: Expand the Equation

Expanding the squared term:

[left(-frac{1}{4}x 40right)^2 left(-frac{1}{4}xright)^2 - 2left(frac{1}{4}xright)40 40^2 ]

Calculating each term:

[left(-frac{1}{4}xright)^2 frac{1}{16}x^2]

[-2left(frac{1}{4}xright)40 -2]

[40^2 1600]

Combining these, we get:

[frac{1}{16}x^2 - 2 1600]

Step 3: Substitute Back into the Circle Equation

Now, substituting back into the circle equation:

[ x^2 left(frac{1}{16}x^2 - 2 1600right) 1850 ]

Combining like terms:

[ x^2 frac{1}{16}x^2 - 2 1600 1850 ]

To combine (x^2) and (frac{1}{16}x^2), convert (x^2) to a fraction:

[ frac{16}{16}x^2 frac{1}{16}x^2 - 2 1600 1850 ]

Combining the (x^2) terms:

[ frac{17}{16}x^2 - 2 1600 1850 ]

Step 4: Simplify the Equation

Subtract 1850 from both sides:

[ frac{17}{16}x^2 - 2 - 250 0 ]

Step 5: Eliminate the Fraction

Multiplying through by 16 to eliminate the fraction:

[ 17x^2 - 32 - 4000 0 ]

Step 6: Solve the Quadratic Equation

The quadratic formula is given by:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Here, a 17, b -320, and c -4000. Calculating the discriminant:

[ b^2 - 4ac (-320)^2 - 4(17)(-4000) 102400 272000 374400 ]

Applying the quadratic formula:

[ x frac{320 pm sqrt{374400}}{34} ]

Calculating (sqrt{374400}):

[ sqrt{374400} approx 612 ]

So, we have:

[ x frac{320 pm 612}{34} ]

Calculating the two possible values for x:

First solution: [ x_1 frac{320 612}{34} frac{932}{34} approx 27.41 ]

Second solution: [ x_2 frac{320 - 612}{34} frac{-292}{34} approx -8.59 ]

Step 7: Determine the Lengths of the Highway

The values of x represent the points where the highway intersects the circle. The usable range for x is from approximately -8.59 to 27.41. Thus:

[ text{Usable Range: From } x approx -8.59 text{ miles to } x approx 27.41 text{ miles.} ]

This gives a total length of:

[ 27.41 - (-8.59) approx 36 text{ miles.} ]

Conclusion

Cars can use their cell phones along approximately 36 miles of the highway. This detailed mathematical approach helps in determining the exact section of a highway that falls within the broadcast signal range of a cellphone tower.