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2 years ago

Graph g(x) = 5|x-6| + 2

Mathematics

1 answer:

Korolek [52]2 years ago

3 0

Answer + Step-by-step explanation:

$f(x) = 5|x-6|+2 = \begin{cases}5\left( x-6\right) +2&if\ x\geq 6\\ 5\left( 6-x\right) +2 &if\ x\leq 6\end{cases}$

$\Longrightarrow f(x) = \begin{cases}5x-30+2&if\ x\geq 6\\ 30-5x +2 &if\ x\leq 6\end{cases}$

$\Longrightarrow f(x) = \begin{cases}5x-28&if\ x\geq 6\\ -5x +32 &if\ x\leq 6\end{cases}$

case 1: x ≥ 6 → f(x) = 5x - 28

5(6) - 28 = 30 - 28 = 2

Then

the point A(6 ,2) lie on the graph (line) of f

5(7) - 28 = 35 - 28 = 7

Then

the point B(7 ,7) lie on the graph (line) of f

Graphing :

When x ≥ 6 ,the graph of f is the ray [AB) (just connect the points A and B)

case 2: x ≤ 6 → f(x) = -5x + 32

-5(6) +32 = -30 + 32 = 2

Then

the point A(6 ,2) lie on the graph (line) of f

-5(5) +32 = -25 + 32 = 7

Then

the point C(5 ,7) lie on the graph (line) of f

Graphing :

When x ≤ 6 ,the graph of f is the ray [AC) (just connect the points A and C)

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3 years ago

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla

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Answer:

150 quartz of creamy vanilla and 50 quartz of Continental Mocha.

Step-by-step explanation:

Let x represents the number of quartz of Creamy vanilla and y represents the number of Continental Mocha,

∵ There is a profit of $ 3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha.

So, the total profit,

Z = 3x + 2y

Which is the objective function for this problem that has to maximise,

Each quart of Creamy Vanilla go 2 eggs and 3 cups of cream,

So, in creamy vanilla,

Number of eggs = 2x

Cups of cream = 3x

Also, Each quart of Continental Mocha go 1 egg and 3 cups of cream,

So, in creamy vanilla,

Number of eggs = y

Cups of cream = 3y

Thus, total number of eggs = 2x + y

And, total number of cups of cream = 3x + 3y

According to the question,

2x + y ≤ 350

3x + 3y ≤ 600

Which are subject of constraint,

Graphing :

Related equation of 2x + y ≤ 350 is 2x + y = 350,

Having x-intercept = (175,0) y-intercept = (0, 350)

Also, 2(0) + 0 ≤ 350 ( True )

∴ Shaded region of 2x + y ≤ 350 would contain the origin,

Related equation of 3x + 3y ≤ 600 is 3x + 3y = 600,

Having x-intercept = (200,0) y-intercept = (0, 200)

Also, 3(0) + 3(0) ≤ 600 ( True )

∴ Shaded region of 3x + 3y ≤ 600 would contain the origin,

'≤' shows solid line,

By graphing them we obtain a feasible region,

In which boundary points are (0, 200), (150, 50) and (175,0)

At (0, 200),

Z = 3(0) + 2(200) = $ 400,

At ( 150, 50)

Z = 3(150) + 2(50) = 450 + 100 = $ 550

At ( 175, 0)

Z = 3(175) + 2(0) = $ 525

Hence, 150 quartz of creamy vanilla and 50 quartz of Continental Mocha should make to earn the largest profit.

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