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emmasim [6.3K]
3 years ago
7

Explain why (4, 1) is not a solution to the equation y=3x+1

Mathematics
1 answer:
nexus9112 [7]3 years ago
6 0

In (4, 1) 4 represents x, and 1 represents y

y = 3x + 1

1 = 3(4)+1

1 = 12 + 1

1 ≠ 13   ← sides are not equal so (4, 1) is not a solution to y=3x+1

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Step-by-step explanation:

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Mashcka [7]

To find the area that is shaded red, you find the area of the rectangle and subtract it by the area of the triangle.


Area of a rectangle:

A = l × w             [ l = 11 in ; w = 6in ]    Plug these numbers into the equation

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Area of a triangle:

A=\frac{1}{2}bh     [ b = 4 in ; h = 6 in ] Plug these #s into the equation

A = \frac{1}{2}(4)(6)

A=\frac{24}{2}

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3 years ago
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zlopas [31]
The vertical distance is 5 + 8 = 13
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Find the slope of the tangent line to the given polar curve at the point specified by the value of θ
MariettaO [177]

The given curve has equation

<em>r(θ)</em> = 9 + 8 cos(<em>θ</em>)

and its derivative is

d<em>r</em>/d<em>θ</em> = -8 sin(<em>θ</em>)

When <em>θ</em> = <em>π</em>/3, we have <em>r</em> (<em>π</em>/3) = 13, and d<em>r</em>/d<em>θ</em> (<em>π</em>/3) = -4√3.

Differentiate these with respect to <em>θ</em> :

d<em>y</em>/d<em>θ</em> = d<em>r</em>/d<em>θ</em> sin(<em>θ</em>) + <em>r(θ)</em> cos(<em>θ</em>)

d<em>x</em>/d<em>θ</em> = d<em>r</em>/d<em>θ</em> cos(<em>θ</em>) - <em>r(θ</em>) sin(<em>θ</em>)

In polar coordinates, we have

<em>y(θ)</em> = <em>r(θ)</em> sin(<em>θ</em>)

<em>x(θ)</em> = <em>r(θ)</em> cos(<em>θ</em>)

and when <em>θ</em> = <em>π</em>/3, we have <em>y</em> (<em>π</em>/3) = 13√3/2 and <em>x</em> (<em>π</em>/3) = 13/2.

The slope of the tangent line to the curve is d<em>y</em>/d<em>x</em>. By the chain rule,

d<em>y</em>/d<em>x</em> = d<em>y</em>/d<em>θ</em> • d<em>θ</em>/d<em>x</em> = (d<em>y</em>/d<em>θ</em>) / (d<em>x</em>/d<em>θ</em>)

d<em>y</em>/d<em>x</em> = (d<em>r</em>/d<em>θ</em> sin(<em>θ</em>) + <em>r(θ)</em> cos(<em>θ</em>)) / (d<em>r</em>/d<em>θ</em> cos(<em>θ</em>) - <em>r(θ</em>) sin(<em>θ</em>))

When <em>θ</em> = <em>π</em>/3, the slope is

d<em>y</em>/d<em>x</em> = (-4√3 sin(<em>π</em>/3) + 13 cos(<em>π</em>/3)) / (-4√3 cos(<em>π</em>/3) - 13 sin(<em>π</em>/3))

d<em>y</em>/d<em>x</em> = (-4√3 (√3/2) + 13 (1/2)) / (-4√3 (1/2) - 13 (√3/2))

d<em>y</em>/d<em>x</em> = - 1/(17√3)

So, the tangent line has slope -1/(17√3) and passes through (13/2, 13√3/2). Using the point-slope formula, its equation is

<em>y</em> - 13√3/2 = -1/(17√3) (<em>x</em> - 13/2)

<em>y</em> = -(<em>x</em> - 338)/(17√3)

4 0
2 years ago
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