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4 days ago

Manuel and Ruben both have bank accounts. The system of equations models their balances after x weeks. y = 11.5x + 22 y = –13x +

218 Their balances will be the same after weeks. Their balances will be $

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Two functions are shown in the table below. Function 1 2 3 4 5 6 f(x) = −x2 + 4x + 12 g(x) = −x + 6 Complete the table on your o

Svet_ta [12734]

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)?

6 0

1 month ago

The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and a standard deviation of 5,00

Svet_ta [12734]

Answer: the likelihood of a randomly selected tire lasting exactly 47,500 miles is 0.067

Step-by-step explanation:

Since the expected lifespan of this tire brand follows a normal distribution, we will use the normal distribution formula:

z = (x - µ)/σ

Where

x = lifespan of the tire in miles.

µ = mean

σ = standard deviation

The given figures include,

µ = 40000 miles

σ = 5000 miles

The probability that a tire will last precisely 47,500 miles

P(x = 47500)

For x = 47500,

z = (40000 - 47500) / 5000 = -1.5

According to the standard normal distribution table, the probability associated with this z score is 0.067

6 0

2 months ago

Island A is 150 miles from island B. A ship captain travels 240 miles from island A and then finds that he is off course and 200

babunello [11817]

Applying the cosine law, we can determine:

c² = a² + b² - 2abcos(C) 
where: 
a, b, and c represent the sides of the triangle and C indicates the angle opposite to side c

Thus, we have:
150² = 240² + 200² - 2(240)(200)cos(C) 

Now we solve for C

22,500 = 57,600 + 40,000 - 96,000cos(C) 
22,500 - 57,600 - 40,000 = -96,000cos(C)
-75,100=-96,000cos (C)
cos (C)=0.7822916
C=arc cos(0.7822916)→ C=38.53°

Therefore, the direction the captain should head towards island B is
180 - 38.53 = 141.47 degrees

4 0

2 months ago

What is the y-intercept of line MN? What is the equation of MN written in standard form?

Leona [12618]

Answer:

The y-intercept for line MN is 2

The standard form of the equation is revealed as ⇒ x + y = 2

Step-by-step elucidation:

Coordinates marking the ends of line MN → M(-3, 5) and N(2, 0)

The slope of the line was computed as $\frac{y_2-y_1}{x_2-x_1}$

= $\frac{5-0}{-3-2}$

= -1

For line MN which passes through (-3, 5) with a slope of -1, the equation formulated is given by

y - 5 = (-1)(x + 3)

This simplifies to

y - 5 = -x - 3

Thus resulting in

y = -x + 2

Here the equation appears in the y-intercept form of

y = mx + b

where m represents the slope of the line and b denotes the y-intercept

So, consequently, the y-intercept for line MN is 2

The equation generates in the standard form as

x + y = 2

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8 0

1 month ago

Read 2 more answers

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Fin

tester [12383]

1 2 3 Step-by-step explanation: Generally, during the roll of two fair 6-sided dice, the doubles result in (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Therefore, the total for doubles is N = 6. The outcome of rolling two fair 6-sided dice yields n = 36. Thus, the probability of rolling doubles (matching numbers on both dice) is calculated mathematically. When rolling two fair dice, outcomes that sum to 4 or less are (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1). Observing this, we see two doubles present. Consequently, the conditional probability of rolling doubles is represented mathematically. Lastly, when rolling the two fair dice, outcomes that show different numbers result in L = 30, while outcomes where at least one die shows a 1 give W = 10. Hence, the conditional probability of having at least one die show a 1 is presented mathematically.

3 0

2 months ago