Answer: the distance between the cars is increasing at 40 mph
Step-by-step explanation:
The direction of movement of both cars forms a right angle triangle. The distance travelled due south and due west by both cars represents the legs of the triangle. Their distance apart after t hours represents the hypotenuse of the right angle triangle.
Let x represent the length the shorter leg(west) of the right angle triangle.
Let y represent the length the longer leg(south) of the right angle triangle.
Let z represent the hypotenuse.
Applying Pythagoras theorem
Hypotenuse² = opposite side² + adjacent side²
Therefore
z² = x² + y²
To determine the rate at which the distances are changing, we would differentiate with respect to t. It becomes
2zdz/dt = 2xdx/dt + 2ydy/dt- - - -- - -1
One travels south at 32 mi/h and the other travels west at 24 mi/h. It means that
dx/dt = 24
dy/dt = 32
Distance = speed × time
Since t = 3 hours, then
x = 24 × 3 = 72 miles
y = 32 × 3 = 96 miles
z² = 72² + 96² = 5184 + 9216
z = √14400
z = 120 miles
Substituting these values into equation 1, it becomes
2 × 120 × dz/dt = (2 × 72 × 24) + (2 × 96 × 32)
240dz/dt = 3456 + 6144
240dz/dt = 9600
dz/dt = 9600/240
dz/dt = 40 mph
Speed is the rate of change of distance over time.
The rate of distance between the cars is increasing at 40 miles per hour.
The distance traveled between both cars, is at right-angled (see attachment).
The distance is calculated as:
[tex]\mathbf{z^2 = y^2 + x^2}[/tex]
Differentiate both sides with respect to time
[tex]\mathbf{2z \frac{dz}{dt} = 2y\frac{dy}{dt} + 2x\frac{dx}{dt} }[/tex]
Divide through by 2
[tex]\mathbf{z \frac{dz}{dt} = y\frac{dy}{dt} + x\frac{dx}{dt} }[/tex]
Where:
[tex]\mathbf{\frac{dx}{dt} = 24mih^{-1}}[/tex]
[tex]\mathbf{\frac{dy}{dt} = 32mih^{-1} }[/tex]
After 3 hours, we have:
[tex]\mathbf{x = 3 \times 24 = 72}[/tex]
[tex]\mathbf{y = 3 \times 32 = 96}[/tex]
So, we have:
[tex]\mathbf{z^2 = y^2 + x^2}[/tex]
[tex]\mathbf{z^2 = 72^2 + 96^2}[/tex]
[tex]\mathbf{z^2 = 14400}[/tex]
Take square roots
[tex]\mathbf{z = 120}[/tex]
[tex]\mathbf{z \frac{dz}{dt} = y\frac{dy}{dt} + x\frac{dx}{dt} }[/tex] becomes
[tex]\mathbf{120 \frac{dz}{dt} = 96 \times 32 + 72 \times 24 }[/tex]
[tex]\mathbf{120 \frac{dz}{dt} = 4800}[/tex]
Divide both sides by 120
[tex]\mathbf{ \frac{dz}{dt} = 40}[/tex]
Hence, the rate of distance between the cars is increasing at 40 miles per hour.
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