Answer:
[tex]\begin{aligned}(1)\quad \textsf{Window washer 1:}\quad h(t)&=4t\\\textsf{Window washer 2:}\quad h(t)&=-6t+50\end{aligned}[/tex]
[tex]\begin{aligned}(2)\quad &\textsf{Time}=5\; \sf hours\\&\textsf{Height}=20\; \sf meters\end{aligned}[/tex]
Step-by-step explanation:
Part (1)
As the window washers ascend and descend at constant rates, we can use linear equations to model their height (h) at any given time (t), where h is measured in meters and t is measured in hours:
[tex]h(t) = mt + b[/tex]
where:
- m is the constant rate they are moving (in meters per hour).
- b is the initial height above ground level (in meters).
The first window washer starts at ground level, so b = 0.
They ascend at a constant rate of 4 meters per hour, so m = 4 (m is positive since they are ascending).
Therefore, the equation that models this window washer's height is:
[tex]\textsf{Window washer 1:}\quad h(t)=4t[/tex]
The second window washer starts at the top of the building at a height of 50 meters above ground level, so b = 50.
They descend at a constant rate of 6 meters per hour, so m = -6 (m is negative since they are descending).
Therefore, the equation that models this window washer's height is:
[tex]\textsf{Window washer 2:}\quad h(t)=-6t+50[/tex]
[tex]\hrulefill[/tex]
Part (2)
To determine the time at which the window washers will be at the same height, and the height at this time, plot the two linear equations on a coordinate plane and find their point of intersection (see attachment).
The x-coordinate of the point of intersection gives the time when the two window washers will be at the same height, and the y-coordinate gives their height at this time.
The point of intersection is (5, 20), which tells us that the window cleaners are at the same height at 5 hours, and that height is 20 meters.
To solve this algebraically, set the height equations equal to each other and solve for t:
[tex]\begin{aligned}4t&=-6t+50\\\\4t+6t&=-6t+50+6t\\\\10t&=50\\\\\dfrac{10t}{10}&=\dfrac{50}{10}\\\\t&=5\; \sf hours\end{aligned}[/tex]
Therefore, the window cleaners will be at the same height after 5 hours.
To determine their height at this time, substitute t = 5 into one of the height equations:
[tex]h(5)&=4(5)[/tex]
[tex]h(5)&=20\; \sf meters[/tex]
Therefore, when the window cleaners are at the same height, that height is 20 meters.
