Question

McMountain Construction is digging a tunnel in the Alps. The mountain can be represented by 3y = 4x + 6000 for 0 ≤ x < 3000 and 7y + 6x = 60000 for 3000 < x < 10000. The tunnel can be represented by the line 10y + x = 30000. All the units are meters. Find the coordinates of the ends of the tunnel.

Answer 1

To find the coordinates of the ends of the tunnel, we need to find the points where the line representing the tunnel intersects with the equations representing the mountain.

Step-by-step explanation:

To find the coordinates of the ends of the tunnel, we need to find the points where the line representing the tunnel intersects with the equations representing the mountain.

For 0 ≤ x < 3000, we can substitute the equation of the tunnel (10y + x = 30000) into the equation of the mountain (3y = 4x + 6000) to solve for y.

For 3000 < x < 10000, we can substitute the equation of the tunnel (10y + x = 30000) into the equation of the mountain (7y + 6x = 60000) to solve for y.

Substitute the tunnel equation (10y + x = 30000) into the second mountain equation (7y + 6x = 60000):

10(30000 - x)/7 = 6x + 60000

428571 - 142857x = 42x + 420000

x = 4714 (this falls within the valid range for x)

Substitute x back into the tunnel equation (10y + x = 30000):

10y + 4714 = 30000

10y = 25286

y = 2528.6

Therefore, the second intersection point is (4714, 2528.6).

Answer 2

The coordinates of the ends of the tunnel are (2000, 1000) and (20000, 1000).

Step-by-step explanation:

The given equations represent different segments of the mountain and the tunnel. For
`\(0 \leq x < 3000\), the equation \(3y = 4x + 6000\)`represents a portion of the mountain. Solving this equation for
`\(y\) gives \(y = (4)/(3)x + 2000\),` indicating the mountain's boundary.

For \(3000 < x < 10000\), the equation
`\(7y + 6x = 60000\)` represents another segment of the mountain. Solving this equation for \(y\) gives
`\(y = -(6)/(7)x + (60000)/(7)\),`signifying the mountain's boundary in this region.

The equation \(10y + x = 30000\) represents the tunnel. Solving for \(y\) gives \(y = \frac{30000 - x}{10}\).

To find the coordinates of the tunnel ends, substitute the equation of the tunnel into the mountain's boundary equations. For the segment where
`\(0 \leq x < 3000\), equate \(y = (4)/(3)x + 2000\) and \(y = (30000 - x)/(10)\), giving \(x = 2000\) and \(y = 1000\). Similarly, for the segment where \(3000 < x < 10000\), equating \(y = -(6)/(7)x + (60000)/(7)\) and \(y = (30000 - x)/(10)\) yields \(x = 20000\) and \(y = 1000\).`

Hence, the tunnel ends at the coordinates (2000, 1000) and (20000, 1000), marking the points where the tunnel intersects the mountain's boundaries.

Understanding how to solve and intersect equations representing boundaries of geographical features, such as mountains and tunnels, is essential for planning construction projects.