Problem 10 Three navy divers are trapped in... [FREE SOLUTION] (2024)

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To understand and solve this problem effectively, creating graphical representations of the equations is essential. By plotting these equations, we visualize how each vehicle's distance to the submarine decreases over time.

The first step is establishing the equations. These are straightforward linear equations, where distance decreases over time as the vehicles move toward the submarine. For example, the helicopter starts 300 miles away and moves at 45 mph, fitting into the equation: \( d_h = 300 - 45h \).

Similarly, for the all-terrain vehicle, it's \( d_a = 130 - 15h \) and for the boat, \( d_b = 100 - 8h \).

By plotting these on a graph with time on the horizontal axis (x-axis) and distance remaining on the vertical axis (y-axis), you see how quickly each vehicle approaches the submarine. At a glance, the graph will show which vehicle reaches zero distance first.

Speed and distance problems are fundamental in algebra because they connect physical concepts with mathematical equations. These types of problems typically involve understanding the relationship between speed, distance, and time. For any object traveling at a constant speed, this relationship is given by the formula:
\[ d = vt \]

Where \( d \) is distance, \( v \) is speed, and \( t \) is time. This concept helps solve the problem by identifying how long it takes each vehicle to reach the submarine.

We started with:

• The helicopter, traveling at 45 mph from 300 miles away
• The all-terrain vehicle, traveling at 15 mph from 130 miles away
• The boat, traveling at 8 mph from 100 miles away

The distances and speeds were then transformed into equations showing how distance changes over time. Solving these equations tells us precisely when each vehicle reaches its goal.

For instance, solving \( 300 - 45h = 0 \) tells us that the helicopter arrives in 6.67 hours. The same logic applies to the other vehicles, making these principles valuable in determining real-life scenarios.

Algebraic equations are key in translating word problems into a format that can be solved mathematically. By defining each vehicle's distance from the submarine as an equation, we break down the problem into manageable parts.

For the helicopter, the equation: \( 300 - 45h = 0 \) tells us when it will arrive. To find the time \( h \), we solve for \( h \):
\[ 300 = 45h \]
\[ h = \frac{300}{45} \]
\[ h = 6.67 \text{ hours} \]

Similarly, for the all-terrain vehicle and the boat:
\[ 130 - 15h = 0 \]
\[ h = \frac{130}{15} \]
\[ h = 8.67 \text{ hours} \]

\[ 100 - 8h = 0 \]
\[ h = \frac{100}{8} \]
\[ h = 12.5 \text{ hours} \]

These steps illustrate how to convert a real-world scenario into algebraic terms, giving precise answers to when each vehicle will reach the submarine. Understanding algebraic concepts like these equips students to handle various mathematical and practical problems, making algebra an indispensable part of the curriculum.