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1 month ago

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A restaurant owner plans to use x tables seating 4, y tables seating 6, and z tables seating 8, for a total of 20 tables. When f

ully occupied, the tables seat 108 customers. If only half of the x tables, half of the y tables, and one-fourth of the z tables are used, each fully occupied, then 46 customers will be seated. Find x, y, and z

1 answer:

babunello [11.8K]1 month ago

5 0

Response:

x = 10, y = 6 and z = 4

Detailed breakdown:

Based on the inquiry, we can establish a set of simultaneous equations featuring 3 variables.

As there are a total of 20 tables, we conclude that

x + y + z = 20

If all seats are filled, x tables for 4 will accommodate 4x customers, y tables for 6 will serve 6y customers, while z tables for 8 will hold 8z customers. We know that if fully occupied, this sums up to 108 customers, leading to the equation

4x + 6y + 8z = 108.

From the last statement "if only half..." we derive another equation:

(4x)/2 + (6y)/2 + (8z)/4 = 46.

Thus, we have these equations (simplifying them yields):

x + y + z = 20 -------------- 1

2x + 3y + 4z = 54 ----------- 2

2x + 3y + 2z = 46 ----------- 3

To solve these equations, subtract equation 3 from equation 2, which gives:

2z = 8

z = 4

Inserting z = 4 into equations 1 and 2

Results in:

x + y = 16 ----------- 4

2x + 3y = 38 -------- 5

To pursue the solution, multiply equation 4 by 2 and maintain equation 5.

4 x 2: 2x + 2y = 32 ----------- 6

5 x 1: 2x + 3y = 38 ------------ 7

Subtract equation 6 from equation 7 yields:

y = 6. Substituting y = 6 and z = 4 back into equation 1 gives x = 10.

As a result, x = 10, y = 6 and z = 4

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For $\fbox{\begin \\\math{x}=6\\\end{minispace}}$ the function $f(x)=-x^{2} +4x+12$ and $g(x)=-x+6$ both yield the same result.

Detailed breakdown:

The functions involved are

$f(x)=-x^{2}+4x+12$

$g(x)=-x+6$

Step 1:

Insert $x=1$ in $f(x)=-x^{2} +4x+12$ to find the value of $f(1)$ .

$f(1)=-1^{2} +4(1)+12\\f(1)=-1+4+12\\f(1)=15$

Insert $x=1$ in $g(x)=-x+6$ to find the value of $g(1)$ .

$g(1)=-1+6\\g(1)=5$

Step 2:

Insert $x=2$ in $f(x)=-x^{2} +4x+12$ to obtain the value of $f(2)$ .

$f(2)=-2^{2} +4(2)+12\\f(2)=-4+8+12\\f(2)=16$

Substitute $x=2$ into $g(x)=-x+6$ to find the value of $g(2)$ .

$g(2)=-2+6\\g(2)=4$

Step 3:

Replace $x=3$ in $f(x)=-x^{2} +4x+12$ to find the value of $f(3)$ .

$f(3)=-3^{2} +4(3)+12\\f(3)=-9+12+12\\f(3)=15$

Also, replace $x=3$ in $g(x)=-x+6$ to find the value of $g(3)$ .

$g(3)=-3+6\\g(3)=3$

Step 4:

Insert $x=4$ in $f(x)=-x^{2} +4x+12$ to find the value of $f(4)$ .

$f(4)=-4^{2} +4(4)+12\\f(4)=-16+16+12\\f(4)=12$

Also, replace $x=4$ in $g(x)=-x+6$ to obtain the value of $g(4)$ .

$g(4)=-4+6\\g(4)=2$

Step 5:

Insert $x=5$ in $f(x)=-x^{2} +4x+12$ to obtain the value of $f(5)$ .

$f(5)=-5^{2} +4(5)+12\\f(5)=-25+20+12\\f(5)=7$

Replace $x=5$ in $g(x)=-x+6$ to find the value of $g(5)$ .

$g(5)=-5+6\\g(5)=1$

Step 6:

Insert $x=6$ into $f(x)=-x^{2} +4x+12$ to find the value of $f(6)$ .

$f(6)=-6^{2} +4(6)+12\\f(6)=-36+24+12\\f(6)=0$

Also, substitute $x=6$ in $g(x)=-x+6$ to obtain the value of $g(6)$ .

$g(6)=-6+6\\g(6)=0$

Step 7:

According to the provided condition $f(x)=g(x)$ .

(a). Insert $f(x)=-x^{2} +4x+12$ and $g(x)=-x+6$ into the previously mentioned equation.

$-x^{2} +4x+12=-x+6$

(b). Multiply through by $-1$ on both sides.

$x^{2} -4x-12=x-6$

(c). Move the term $x-6$ to the left side of the equation.

$x^{2} -4x-12-x+6=0\\x^{2} -5x-6=0$

(d). Divide the middle term so that its sum equals 5 and the product equals 6.

$x^{2} -(6-1)x-6=0\\x^{2} -6x+x-6=0\\x(x-6)+1(x-6)=0\\(x+1)(x-6)=0\\x=-1,6$

From the analysis above, it is noted that for $x=6$ both functions $f(x)$ and $g(x)$ yield the same outcome.

Using a direct approach:

$f(x)=g(x)\\\Leftrightarrow-x^{2} +4x+12=-x+6\\\Leftrightarrow-x^{2} +4x+12+x-6=0\\\Leftrightarrow-x^{2} +5x+6=0\\\Leftrightarrow-x^{2} +6x-x+6=0\\\Leftrightarrow x^{2} -6x+x-6=0\\\Leftrightarrow x(x-6)+1(x-6)=0\\\Leftrightarrow(x+1)(x-6)=0\\\Leftrightarrow x=6,-1$

The table representing function $f(x)=-x^{2} +4x+12$ and $g(x)=-x+6$ is included below.

For more information:

1. What is the y-intercept of the quadratic function f(x) = (x – 6)(x – 2)? (0,–6) (0,12) (–8,0) (2,0)

2. Which is the graph of f(x) = (x – 1)(x + 4)?

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Answer:

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Detailed solution:

Problem statement:

Determine the volume of the open-topped box as a function of the side length x (in centimeters) of the square cutouts.

Refer to the provided diagram for clarity.

Define:

x → length in centimeters of each square cutout side

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$V=LWH$

Given this, we have:

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By substitution:

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3 months ago

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tester [12383]

Answer:

Attached is the question in consideration.

$m\angle CEA =90 \ (deg)$

$m\angle BEF=135\ (deg)$

$\angle CEF$ forms a straight line.

$\angle AEF$ depicts a right triangle.

The options $1,4,5,6$ represent the correct answers.

Step-by-step explanation:

⇒ Given that $\ ray\ AE$ is ⊥ $FEC$ it constitutes a right triangle, leading to $m\angle CEA =90\ (deg)$ .

⇒ The measure for $\angle BEF =135\ (deg)$ equals $\angle BEF =\angle AEB +\angle AEF = (45+90)=135\ (deg)$ as $\angle AEB$ bisects $\angle AEC$ , implying that $\angle AEB$ is half of $\angle AEC$ , thus $\angle AEB = 45\ (deg)$ .

⇒ $\angle CEF$ represents a straight line, as the angle measures across it yield $180\ (deg)$ .

⇒ The angle measure for $\angle AEF = 90\ (deg)$ is derived from the linear pair concept.

Since $\angle CEA + \angle AEF = 180\ (deg)$ , inserting the values of $m\angle CEA =90\ (deg)$ leads to $\angle AEF = 90\ (deg)$ .

The other two options are incorrect as:

$m\angle CEF=m\angle CEA + m\angle BEF = (90+135)=225$

it surpasses $180\ (deg)$ while $\angle CEF$ is a

straight line.

Also, $m\angle CEB=2(m\angle CEA)$ is inaccurate.

As $\angle CEA = 90\ (deg)$ and $\angle CEB=45\ (deg)$

Thus, we have a total $4$ valid answers.

The confirmed options are $1,4,5,6$ .

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