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four cities lie along a straight highway in this order city a, city, city c, city d. The distance from city a to city b is 5 times the distance from city b to city c. The distance from city a to city d is 2 times the distance from city a to city d. the distance between city d and city c is 10 miles. What are the distances between the cities?

Respuesta :

Dividing the segments and finding the lengths, it is found that the distance between the two cities is of 130 miles.

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  • We want to find the distance between cities a and d.
  • Due to the cities along the highway, the distance is given by:

[tex]ad = ab + bc + cd[/tex]

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  • The distance from city a to city b is 5 times the distance from city b to city c, and thus:

[tex]ab = 5bc[/tex]

In terms of x, [tex]bc = x, ab = 5x[/tex]

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  • The distance from city b to city c is 2 times the distance from city c to city d, and thus:

[tex]bc = 2cd[/tex]

In terms of x, [tex]cd = \frac{x}{2}[/tex]

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  • The distance between city d and city c is 10 miles, thus:

[tex]\frac{x}{2} = 10[/tex]

[tex]x = 20[/tex]

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  • The other distances are:

[tex]bc = x = 20[/tex]

[tex]ab = 5x = 5(20) = 100[/tex]

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  • The distance between cities a and d is:

[tex]ad = ab + bc + cd = 100 + 20 + 10 = 130[/tex]

The distance between the two cities is of 130 miles.

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