Find all values of k so that the following system has no solutions x²+(y-k)²=36 ; y=5x+k.Illustrate with a graph.

Mathematics

1 answer:

musickatia [10]2 years ago

5 0

Step-by-step explanation:

$x^2+(y-k)^2=36$

$y=5x+k$

Substituting y in the first equation

$x^2+(5x+k-k)^2=36\\\Rightarrow x^2+25x^2=36\\\Rightarrow 26x^2=36\\\Rightarrow x^2=\frac{36}{26}\\\Rightarrow x=\sqrt{\frac{36}{26}}\\\Rightarrow x=\pm 1.17$

At x = 1.17

$y=5\times 1.17+k\\\Rightarrow y=5.85+k$

At x = -1.17

$y=5\times -1.17+k\\\Rightarrow y=-5.85+k$

The intersection points are (1.17, 5.85k) and (-1.17, -5.85k)

The graph would intersect regardless of what value k is.

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Lily swims for 5 hours in stream that has a current of 1 mph. She leaves her dock, swims upstream for 2 miles and then swim back

antiseptic1488 [7]

Answer:

Swimming speed in still water is approximately 1.5 mph

Step-by-step explanation:

Given the speed of the current = 1 mph

distance she swims upstream =2 miles,

the total time = 5 hours.

She swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is $d = r t\ \ \ \ or \ \ \ t= \frac{d}{r}$

Let x be the speed in still water.

Then her speed with the current is x + 1, and

her speed against the current is x – 1.

Total time is equal to 2 miles with upstream and 2 miles downstream.

$\frac{2}{x+1}+\frac{2}{x-1}=5\\\\\frac{2(x-1)}{(x+1)(x-1)}+\frac{2(x+1)}{(x-1)(x+1)}=5\\\\\frac{2x-2}{(x^2-1)}+\frac{2x+2)}{(x^2-1)}=5\\\\\frac{2x-2+2x+2)}{(x^2-1)}=5\\\\4x=5(x^2-1)\\4x= 5x^2-5\\0= 5x^2-4x-5$

Now Using quadratic formula to solve above equation we get;

$x= \frac{-b \±\sqrt{b^2-4ac}} {2a}\\\\here \ \ a= 5,b=-4,c=-5\\\\x= \frac{-(-4) \±\sqrt{(-4)^2-4\times 5\times-5}} {2\times5}\\\\x=\frac{4 \±\sqrt{16+100}} {10}\\\\x=\frac{4 \±\sqrt{116}} {10}=x=\frac{4 \±2\sqrt{29}} {10}= x=\frac{2 \±\sqrt{29}} {5}\\ x \approx 1.5 \ or \ x = -0.7$