To answer question 12, let’s set up the system of equations for the situation:

1. Define variables:

   • Let $t$ be the time in minutes.
   • Let $h_M(t)$ and $h_D(t)$ be the altitudes of Mohamed's and Dana's balloons at time $t$, respectively.

2. Equation for Mohamed's balloon:

   • Starting altitude: 40 m
   • Rising rate: 10 m/min
   • Equation: $h_M(t) = 40 + 10t$

3. Equation for Dana's balloon:

   • Starting altitude: 165 m
   • Descending rate: -15 m/min
   • Equation: $h_D(t) = 165 - 15t$

Part (a)

To find when both balloons are at the same altitude, set $h_M(t) = h_D(t)$:

$40 + 10t = 165 - 15t$

Solve for $t$:

1. Combine terms: $25t = 125$
2. Divide by 25: $t = 5$

So, both balloons will be at the same altitude after 5 minutes.

Part (b)

To find the altitude at this time, substitute $t = 5$ into either equation:

Using $h_M(t) = 40 + 10t$: $h_M(5) = 40 + 10(5) = 40 + 50 = 90 \text{ m}$

Thus, the altitude will be 90 meters.

Question 13 Solution

For the next problem:

• Define variables for ages:

  • Let $x$ be Teo's age.
  • Richard's age would then be $2x + 4$.

• Given that their combined age is 31: $x + (2x + 4) = 31$

  Simplify and solve: $3x + 4 = 31$ $3x = 27$ $x = 9$

  So, Teo is 9 years old, and Richard is: $2(9) + 4 = 18 + 4 = 22 \text{ years old}$

Would you like a more detailed explanation, or have further questions?

Additional Questions for Practice:

1. What if Mohamed's balloon started at a different altitude? How would that change the equations?
2. How would the answer change if Dana's balloon descended at a rate of 10 m/min instead of 15 m/min?
3. If both balloons were rising, how would the system of equations differ?
4. How would you approach solving for age if Richard was 3 years older than twice Teo’s age instead of 4?
5. What are the steps to verify the answers in both problems?

Tip:

Always start by defining variables and setting up clear equations for each condition in a word problem.