Mathematics, 07.04.2020 16:59 serenityarts123

Two cars start moving from the same point. One travels south at 32 mi/h and the other travels west at 24 mi/h. At what rate is the distance between the cars increasing three hours later

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Answer from: samsmith666

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:

$\mathbf{z^2 = y^2 + x^2}$

Differentiate both sides with respect to time

$\mathbf{2z \frac{dz}{dt} = 2y\frac{dy}{dt} + 2x\frac{dx}{dt} }$

Divide through by 2

$\mathbf{z \frac{dz}{dt} = y\frac{dy}{dt} + x\frac{dx}{dt} }$

Where:

$\mathbf{\frac{dx}{dt} = 24mih^{-1}}$

$\mathbf{\frac{dy}{dt} = 32mih^{-1} }$

After 3 hours, we have:

$\mathbf{x = 3 \times 24 = 72}$

$\mathbf{y = 3 \times 32 = 96}$

So, we have:

$\mathbf{z^2 = y^2 + x^2}$

$\mathbf{z^2 = 72^2 + 96^2}$

$\mathbf{z^2 = 14400}$

Take square roots

$\mathbf{z = 120}$

$\mathbf{z \frac{dz}{dt} = y\frac{dy}{dt} + x\frac{dx}{dt} }$ becomes

$\mathbf{120 \frac{dz}{dt} = 96 \times 32 + 72 \times 24 }$

$\mathbf{120 \frac{dz}{dt} = 4800}$

Divide both sides by 120

$\mathbf{ \frac{dz}{dt} = 40}$

Hence, the rate of distance between the cars is increasing at 40 miles per hour.

Read more about distance and rates at:

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Answer from: jeanlucceltrick09

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

Two cars start moving from the same point. One travels south at 32 mi/h and the other travels west a

Answer from: Quest

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

Answer from: Quest

yes the total will be same because in both of them the expression.is rearranged but their sign remains the same hence it hss just been shuffled

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Two cars start moving from the same point. One travels south at 32 mi/h and the other travels west a...