Let's go through the problems in the image one by one.

Problem 5

Statement: Find the length of the road connecting Kecamatan C and Kecamatan H.

Solution: From the image, it appears that the path connecting C and H is a straight line, which can be interpreted as the hypotenuse of a right triangle. If the distance $CD = 40 \, \text{km}$ is given and knowing that $DE = 15 \, \text{km}$, we can use the Pythagorean theorem to find the length $CH$.

Let's denote:

• $CD = 40 \, \text{km}$
• $DH = \sqrt{(CD)^2 + (DE)^2}$
• $DE = 15 \, \text{km}$

Plugging in the values: $CH = \sqrt{(40)^2 + (15)^2} = \sqrt{1600 + 225} = \sqrt{1825} = 43 \, \text{km}$

This matches with option E.

Problem 6

Statement: Find the shortest distance from Kecamatan C to Kecamatan E by car.

Solution: The shortest distance will follow the straight path from C to E, since the other path involves a curved route.

Given that the coordinate plane distance between $C$ and $E$ is equivalent to moving diagonally across a right triangle formed by distances:

• $CE = \sqrt{(15)^2 + (15)^2} = \sqrt{450} = 15\sqrt{2} \, \text{km}$

This matches with option C.

Problem 7

Statement: Calculate the time Andre needs to travel from Kecamatan G to D, passing through E and F.

Solution:

1. Total Distance Calculation:

   • $GE$ is a quarter circle.
   • $EF$ is a straight line.
   • $FD$ is another straight line.

2. Speed Calculation:

   • Average speed on the road is 50 km/h.
   • Speed on the toll road is 30 km/h.

3. Time Calculation: This requires more detailed information on the specific lengths which needs to be extracted from the figure.

Problem 8

Statement: Determine when Hadi should start driving from Kecamatan G to meet Badu at F exactly when Badu arrives.

Solution:

1. Speed Comparison:

   • Speed of Hadi: $30 \, \text{km/h}$.
   • Speed of Badu: $15\sqrt{5} \, \text{km/h}$.

2. Distance and Time Synchronization:

   • Badu starts at 08:00 AM.
   • Calculate the meeting time at F.

We'd need to set up equations using the distances and speeds to find the exact starting time for Hadi.

Would you like detailed calculations for all the problems above, or do you have specific ones you’d like me to explain further?

Further Questions

1. How do we calculate the time taken on different types of roads considering speed limits?
2. What would be the formula for calculating the time Andre needs for the journey if he changes speeds at different segments?
3. How can we use trigonometric identities to find distances between points in the coordinate plane?
4. How do you compute the distance of curved paths like quarter circles in coordinate geometry?
5. What is the role of the Pythagorean theorem in solving distance problems in coordinate planes?

Tip:

Always break down the problem into smaller segments and solve step-by-step to avoid confusion, especially when dealing with mixed types of paths like straight lines and curves.