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2 years ago

Find the distance d (P1,P2) between the given points P1 and P2 P1= (5,4) P2=(-4,5)

Mathematics

1 answer:

oee [108]2 years ago

5 0

Answer:

$\sqrt{82 }$

Step-by-step explanation:

d = $\sqrt{(x_{1} - x_2)^2+(y_1-y_2)^2 }$

d = $\sqrt{(5 - (-4))^2+(4-5)^2 }$

d = $\sqrt{(9)^2+(-1)^2 }$

d = $\sqrt{81+1 }$ = $\sqrt{82 }$

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You are a lifeguard and spot a drowning child 60 meters along the shore and 40 meters from the shore to the child. You run along

sukhopar [10]

Answer:

The lifeguard should run across the shore a distance of 48.074 m before jumpng into the water in order to minimize the time to reach the child.

Step-by-step explanation:

This is a problem of optimization.

We have to minimize the time it takes for the lifeguard to reach the child.

The time can be calculated by dividing the distance by the speed for each section.

The distance in the shore and in the water depends on when the lifeguard gets in the water. We use the variable x to model this, as seen in the picture attached.

Then, the distance in the shore is d_b=x and the distance swimming can be calculated using the Pithagorean theorem:

$d_s^2=(60-x)^2+40^2=60^2-120x+x^2+40^2=x^2-120x+5200\\\\d_s=\sqrt{x^2-120x+5200}$

Then, the time (speed divided by distance) is:

$t=d_b/v_b+d_s/v_s\\\\t=x/4+\sqrt{x^2-120x+5200}/1.1$

To optimize this function we have to derive and equal to zero:

$\dfrac{dt}{dx}=\dfrac{1}{4}+\dfrac{1}{1.1}(\dfrac{1}{2})\dfrac{2x-120}{\sqrt{x^2-120x+5200}} \\\\\\\dfrac{dt}{dx}=\dfrac{1}{4} +\dfrac{1}{1.1} \dfrac{x-60}{\sqrt{x^2-120x+5200}} =0\\\\\\ \dfrac{x-60}{\sqrt{x^2-120x+5200}} =\dfrac{1.1}{4}=\dfrac{2}{7}\\\\\\ x-60=\dfrac{2}{7}\sqrt{x^2-120x+5200}\\\\\\(x-60)^2=\dfrac{2^2}{7^2}(x^2-120x+5200)\\\\\\(x-60)^2=\dfrac{4}{49}[(x-60)^2+40^2]\\\\\\(1-4/49)(x-60)^2=4*40^2/49=6400/49\\\\(45/49)(x-60)^2=6400/49\\\\45(x-60)^2=6400\\\\$

$x$

As $d_b=x$ , the lifeguard should run across the shore a distance of 48.074 m before jumpng into the water in order to minimize the time to reach the child.

7 0

2 years ago

What is the equation of the line in the graph

iren [92.7K]

Always, ALWAYS remeber this format: y = mx + b

In this equation, 'm' is the slope, and 'b' is the y-intercept

When you're trying to find a slope, remember that the equation is $\frac{rise}{run}$

When finding the rise and run, look at two points that are on the graph AND on the line as well. Essentially, make sure the points you're using are integers.

In this, case, the rise is -3, and the run is 2. This means that the slope is $\frac{-3}{2}$

Now we have the first part of our equation:

y = - $\frac{3}{2}$ + b

But wait! How do we find b?

Sometimes you have to input x in order to find it, but only when you're not supplied with a graph. In this case, all you have to do is look!

The point of the line that is on the y-axis is called the y-intercept.

In this graph, the y-intercept is -1

Now we have our complete equation!

y = - $\frac{3}{2}$ - 1

Good luck!

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