Answer:<em> 10 miles</em>
Step-by-step explanation:
<em>Mile 1: 3.00</em>
<em>Mile 2: 3.50</em>
<em>Mile 3: 4.00</em>
<em>Mile 4: 4.50</em>
<em>Mile 5: 5.00</em>
<em>Mile 6: 5.50</em>
<em>Mile 7: 6.00</em>
<em>Mile 8: 6.50</em>
<em>Mile 9: 7.00</em>
<em>Mile 10: 7.50</em>
The longest possible altitude of the third altitude (if it is a positive integer) is 83.
According to statement
Let h is the length of third altitude
Let a, b, and c be the sides corresponding to the altitudes of length 12, 14, and h.
From Area of triangle
A = 1/2*B*H
Substitute the values in it
A = 1/2*a*12
a = 2A / 12 -(1)
Then
A = 1/2*b*14
b = 2A / 14 -(2)
Then
A = 1/2*c*h
c = 2A / h -(3)
Now, we will use the triangle inequalities:
2A/12 < 2A/14 + 2A/h
Solve it and get
h<84
2A/14 < 2A/12 + 2A/h
Solve it and get
h > -84
2A/h < 2A/12 + 2A/14
Solve it and get
h > 6.46
From all the three inequalities we get:
6.46<h<84
So, the longest possible altitude of the third altitude (if it is a positive integer) is 83.
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