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Rasek [7]
3 years ago
13

Please help im confused\

Mathematics
1 answer:
Bad White [126]3 years ago
8 0

The slope-point form: y - y₁ = m(x - x₁).

The slope-intercept form: y = mx + b.

k:y=m_1x+b_1,\ l:y=m_2x+b_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}

We have:

k:y-3=-4(x+2)\to m_1=-4 and the point (-5, 7)

therefore m_2=-\dfrac{1}{-4}=\dfrac{1}{4}

y-7=\dfrac{1}{4}(x-(-5))\\\\\boxed{y-7=\dfrac{1}{4}(x+5)}\to\boxed{B.}

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Feliz [49]

The base length that will maximize the area for such a window is 168.03 cm. The exact largest value of x when this occurs is 233.39 cm

Suppose we make an assumption that:

  • (x) should be the width of the rectangle base;
  • (h) should be the height of the rectangle

Also, provided that the diameter of the semi-circle appears to be the base of the rectangle, then;

  • the radius  \mathbf{r = \dfrac{x}{2}}  

and, the perimeter of the window can now be expressed as:

\mathbf{x + 2h + \pi r = x + 2h + \dfrac{\pi x }{2}}

\mathbf{= \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h}

Given that the perimeter = 600 cm

∴

\mathbf{ \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h= 600}

\mathbf{  h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x}

Since h > 0, then:

\mathbf{  h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x>0}

By rearrangement and using the inverse rule:

\mathbf{  x<  \dfrac{ 300}{\Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) } }

\mathbf{  x=  \dfrac{ 1200}{\Big( 2 +\pi \Big) } }

\mathbf{  x=  233.39 \ cm }

Thus, the largest length x = 233.39 cm

However, the area of the window is given as:

\mathbf{A(x) = xh + \dfrac{1}{2} \pi r^2}

\mathbf{A = x \Big [  300 - \Big ( \dfrac{1}{2}+\dfrac{1}{4} \Big) x \Big ]  +\dfrac{1}{2}\pi \Big(\dfrac{x}{2} \Big )^2}

\mathbf{A (x) = 300x - \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ cm^2}

Now, at maximum, when the area A = 0. Taking the differentiation, we have:

\mathbf{\dfrac{d}{dx} 300x - \dfrac{d}{dx} \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ =0}

\mathbf{ 300 - 2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big)  \ =0}

Making x the subject of the formula, we have:

\mathbf{x = \dfrac{1200}{4 +\pi}}

x = 168.03 cm

Taking the second derivative:

\mathbf{\dfrac{d}{dx} \Big [300 -2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \Big]}

\mathbf{= -2 \Big( \dfrac{1}{2}+\dfrac{\pi}{8}\Big )

Therefore, we can conclude that the maximum area that exists for such a window is 168.03 cm

Learn more about derivative here:

brainly.com/question/9964510?referrer=searchResults

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you would switch the 2nd one so the 10 would be on top.

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