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3 years ago

Write the equation of a line that is parallel to 3x+2y=10 and passes through the point (4,-5). Answer in slope-intercept form.

Mathematics

1 answer:

SCORPION-xisa [38]3 years ago

4 0

The equation of line is:

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

Further explanation:

The standard form of equation in point-slope form is:

$y = mx+b$

Given equation is:

3x+2y=10

We have to convert it into point slope form, for which we have to isolate y on one side of equation

$3x+2y = 10\\2y = -3x +10\\y = \frac{-3}{2}x + \frac{10}{2}\\y = \frac{-3}{2}x + 5$

Comparing with standard form:

m = -3/2

As the new line is parallel to given line, their slopes will be equal

So,

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

To find the value of b, putting the given point (4,-5) in equation

$-5 = \frac{-3}{2}(4)+b\\-5 = -6+b\\b = -5+6 \\b = 1\\So\ the\ equation\ is:\\y = \frac{-3}{2}x+1$

Keywords: Point-slope form, equation of line

Learn more about point-slope form of equation at:

#LearnwithBrainly

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Answer:

Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.

Therefore the minimum cost to manufacture the box is $12936.

Step-by-step explanation:

Given the volume of the open box is 196 cube ft.

Let the one side of the base of the box be x.

Since the base of the box is square in shape.

The other side of the base is also x.

The area of the base of the box = x² square ft

Let the height of the box be h.

The volume of the box is = Area of the base× height

=(x²×h) cube ft.

=x²h cube ft.

According to the problem,

x²h=196

$\Rightarrow h=\frac{196}{x^2}$

The material cost for the bottom is $88 per square ft and the material cost for the sides $77 per square ft.

The material cost for the bottom of the box = $(88×x²)

=$88x²

The area sides = 2(length+width)height

=2(x+x)h

=4xh square ft.

The material cost for the sides of the box =$(77×4xh)

=$308 xh

Total cost to make the box is =$(88x²+308 xh)

∴C=88x²+308 xh [ C is in dollar]

Now putting $h=\frac{196}{x^2}$

$C=88x^2+308x\times \frac{196}{x^2}$

$\Rightarrow C=88x^2+ \frac{60368}{x}$

Differentiating with respect to x

$C'=176x- \frac{60368}{x^2}$

Again differentiating with respect to x

$C''=176+ \frac{120736}{x^3}$

Now to find the minimum cost, we set C'=0

$\therefore 176x- \frac{60368}{x^2}=0$

$\Rightarrow 176x= \frac{60368}{x^2}$

$\Rightarrow x^3= \frac{60368}{176}$

$\Rightarrow x^3=343$

$\Rightarrow x=7$

Now, $C''|_{x=7}=176+ \frac{120736}{7^3}>0$

Therefore at x=7, the manufacturing cost will be minimum.

The height of the box is $h=\frac{196}{7^2}$ = 4ft

Therefore the dimensions of the box is 7 ft by 7 ft by 4 ft.

The manufacturing cost is $C=88.7^2+ \frac{60368}{7}$ [ putting x=7]

=$12936

Therefore the minimum cost to manufacture the box is $12936.

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3 years ago