Find the Equation of the Perpendicular Line
Instructions: Find the equation of the line through point (-1, 2) and perpendicular to x + 3y = 3.

y =

Take any two pairs of the given points and make vectors out of them. For example, the vector from A to B is

[tex]\vec v_1 = \langle 1,4,5\rangle - \langle 1,1,1\rangle = \langle0,3,4\rangle[/tex]

and the vector from A to C is

[tex]\vec v_2 = \langle -3,-2,0\rangle - \langle1,1,1\rangle = \langle-4,-3,-1\rangle[/tex]

These vectors lie in the same plane (the one we want). We can get a third vector that is normal to the plane by taking their cross product (details omitted).

[tex]\vec n = \vec v_1 \times \vec v_2 = \langle 9,-16,12 \rangle[/tex]

If [tex]\vec u = \langle x,y,z\rangle[/tex] is an arbitrary vector, then the vector from any of the points A, B, or C to [tex]\vec u[/tex] will lie in our plane. That is, if we start from A,

[tex](\vec u - \langle1,1,1\rangle) \cdot \vec n = 0[/tex]

and this reduces to the equation of the plane,

[tex]\langle x - 1, y - 1, z - 1 \rangle \cdot \langle 9, -16, 12 \rangle = 0[/tex]

[tex]9 (x - 1) - 16 (y - 1) + 12 (z - 1) = 0[/tex]

[tex]\boxed{9x - 16y + 12z = 5}[/tex]

This follows immediately from the cross product identity

[tex]\|\vec x \times \vec y\| = \|\vec x\| \|\vec y\| \sin(\theta)[/tex]

where [tex]\theta[/tex] is the angle between [tex]\vec x[/tex] and [tex]\vec y[/tex]. The left side corresponds to the area of the parallelogram spanned by [tex]\vec x[/tex] and [tex]\vec y[/tex]; half of this area would be that of a triangle. (see attached)

In our case, we have

[tex]\|\vec n\| = \|\vec v_1 \times \vec v_2\| = \sqrt{481}[/tex]

so the area of ABC is [tex]\boxed{\dfrac{\sqrt{481}}2}[/tex].

We already know the first vector, [tex]v_1[/tex].

The vector from C to B is

[tex]\vec v_3 = \langle 1,4,5 \rangle - \langle -3,-2,0 \rangle = \langle 4, 6, 5 \rangle[/tex]

Recall the dot product identity,

[tex]\vec x \cdot \vec y = \|\vec x\| \|\vec y\| \cos(\theta)[/tex]

Then

[tex]\vec v_1 \cdot \vec v_3 = \|\vec v_1\| \|\vec v_3\| \cos(\theta) \implies \cos(\theta) = \dfrac{38}{5\sqrt{77}} \implies \theta \approx \boxed{29.9914^\circ}[/tex]

Answer:

Step-by-step explanation:

cos (x/2)=cos x+1

cos (x/2)=2cos ²(x/2)

2 cos²(x/2)-cos (x/2)=0

cos (x/2)[2 cos (x/2)-1]=0

cos (x/2)=0=cos π/2,cos (3π/2)=cos (2nπ+π/2),cos(2nπ+3π/2)

x/2=2nπ+π/2,2nπ+3π/2

x=4nπ+π,4nπ+3π

n=0,1,2,...

x=π,3π

or x=180°,540°,...

180°∈[0,360]

so x=180°

or

2cos(x/2)-1=0

cos (x/2)=1/2=cos60,cos (360-60)=cos 60,cos 300=cos (360n+60),cos (360n+300)

x/2=360n+60,360n+300

x=720n+120,720n+300

n=0,1,2,...

x=120,300,840,1020,...

only 120° and 300° ∈[0,360°]

Hence x=120°,180°,300°

The amount of the radioactive substance is 374.6 g

The given parameters are:

The amount of the radioactive substance is calculated as:

A(t) = a(1 - r)^t

This gives

A(t) = 424 * (1 - 6%)^t

At 2 hours, we have:

A(2) = 424 * (1 - 6%)^2

Evaluate

A(2) = 374.6

Hence, the amount of the radioactive substance is 374.6 g

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The area of the shaded region for a z-score of -0.92 is 0.1788, which means the area that will be shadeded in the normal distribution graph is 17.88% of the total.

The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution about the mean, indicating that data near the mean occur more frequently than data distant from the mean. The normal distribution will show as a bell curve on a graph.

1.) The area of the shaded region for a z-score of -0.92 is 0.1788, which means the area that will be shadeded in the normal distribution graph is 17.88% of the total.

2.) The area of the shaded region for a z-score of 1.23 is 0.8907, which means the area that will be shadeded in the normal distribution graph is 89.07% of the total.

Now, the area of the shaded region will be,

Area = 0.8907 - 0.1788

        = 0.7119

        = 71.19%

Hence, the area of the shaded region is 71.19%.

The complete question is mentioned in the below image.

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The equation that represents the given problem would be (n-30) + n = 210. The correct option is the option A; (n-30) + n = 210

A linear equation is an equation that has the variable of the highest power of 1.

The standard form of a linear equation is of the form Ax + B = 0.

From the given information,

n = Tom's weight in pounds

If Jim weighs 30 pounds less than Tom

Jim weighs will be (n - 30)

Thus,

The sum of their weight

(n-30) + n

∴ (n-30) + n = 210

Hence, the equation that represents the given problem would be (n-30) + n = 210. The correct option is the option A; (n-30) + n = 210

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Answer:

x = 4

Step-by-step explanation:

Given functions:

[tex]\begin{cases}f(x)=3x-1\\g(x)=2x-3\end{cases}[/tex]

f(2) means substitute x = 2 into function f(x):

[tex]\begin{aligned}g(x) & = f(2)\\ \implies 2x-3 & = 3(2)-1\\2x-3 & = 6-1\\2x - 3 & = 5\\2x - 3 + 3 & = 5 + 3\\2x & = 8\\\dfrac{2x}{2} & = \dfrac{8}{2}\\x & = 4\end{aligned}[/tex]

Therefore,  g(4) = f(2).

f(2)

so

⊰_________________________________________________________⊱

Answer:

Step-by-step explanation:

[tex]\large\displaystyle\text{$\begin{gathered} \sf{Substitute \ the \ values \ into \ the \ formula \ y-y_1=m(x-x_1)} \\ \sf {parallel \ lines \ have \ same \ slopes, \ thus} \\ \sf{slope \ of \ the \ 2nd \ line = 3}\\ \sf{now \ substitute \ the \ values} \\ \sf {y-(-5)=3(x-4)}\\ \sf{y+5=3(x-4) (It's \ Point-Slope\;Form, \ see \ below \ for \ slope-intercept)}\\ \sf {y+5=3x-12} \\ \sf{y=3x-12-5} \\ \sf{y-3x-17} \end{gathered}$}}[/tex]

[tex]\pmb{\tt{done \ !!}}[/tex]

⊱_________________________________________________________⊰

a. The probability of rolling a sum of 7 or 11. (sum of 7 or 11) is 2/9. b. the probability of not rolling a sum of 7 nor 11. (not sum of 7 nor 11) is 7/9.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

The probability of getting a total of 7 = 6/36

The probability of getting total of 11 = 2/36

a. The probability of rolling a sum of 7 or 11. (sum of 7 or 11)

= 6/36 + 2/36

= 2/9

b. Find the probability of not rolling a sum of 7 nor 11. (not sum of 7 nor 11)

= 1- 2/ 9

= 7/9

c. Find the odds against rolling a sum of 7 or 11.

= 2/9

d. The money you will win, If you make a $10 bet that you will roll a sum of 7 or 11, and the dice land on a sum of 7 or 11.

10 x  2/9 = 20/9

10 x 7/9 = 70/9

Thus, The money you will win $10.

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Answer:

Your answer is 71

Step-by-step explanation:

7+2(n-1)

we have our n as 8

so,7+2(8-1)

9(7)=63

Answer:

$100

Step-by-step explanation:

Given information:

Let x = money taken from bank account

If Sally paid $54 for her ticket, then the money remaining at that point can be defined as:  [tex]x-54[/tex]

If the cost of the hotel was half of the remaining money, then:

[tex]\textsf{Cost of hotel}=\dfrac{1}{2}(x-54)}[/tex]

To find how much money Sally took from her bank account, create an equation with the given information and solve for x.

[tex]\implies \textsf{Money in bank} - \textsf{ticket} - \textsf{hotel} - \textsf{ food} =\textsf{remaining money}[/tex]

[tex]\implies x - 54 - \dfrac{1}{2}(x - 54)-7.90=15.10[/tex]

[tex]\implies x - 54 - \dfrac{1}{2}x+27-7.90=15.10[/tex]

[tex]\implies \dfrac{1}{2}x-34.90=15.10[/tex]

[tex]\implies \dfrac{1}{2}x=50[/tex]

[tex]\implies x=100[/tex]

Therefore, Sally took $100 from her bank account.

Draw the graph:

f(x) = |x-1|

f(x) = -(x-1) when x-1 is negative

f(x) = (x-1) when x-1 is positive

when x = -2, f(x) = 3

when x = -1, f(x) = 2

when x = 0, f(x) = 1

when x = 1, f(x) = 0

when x = 2, f(x) = 1

Using this values, draw the graph of f(x) = |x-1|

Hence, the required modulus of x-1 graph.

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Answer:

125

Step-by-step explanation:

N150=120%

? =100

150×100/120

=125

The cost price of the good is $125.

Cost price is the total amount of money that it costs a manufacturer to produce a given product or provide a given service.

Given that a good has been sold for $150 which made a profit of 20% we need to find the cost price of the good,

Let the cost price be x,

So,

x × 120% = 150

x = 150 × 100 / 120

x = 125

Hence the cost price of the good is $125.

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Answer:

Width: 6

Length: 20

Step-by-step explanation:

So the area of a rectangle can be defined as: [tex]A=wl[/tex] where w=width and l=length.

In this case we don't know what the length is, so let's just say the length is the variable l, and since the width is 14 units less than the length, we can express it as (l-14). this gives us the equation: [tex]A=l(l-14)=l^2-14l[/tex]. We can solve for l, since we're given the area which is 120. So let's set the equation equal to that:

Original Equation:

[tex]A=l^2-14l[/tex]

Substitute 120 as A (given)

[tex]120=l^2-14l[/tex]

There is many ways to solve this equation: factoring, quadratic equation, completing the square etc... but in this case I'll just complete the square

Add (b/2)^2 to both sides to complete the square

[tex]120+(\frac{-14}{2})^2=l^2-14l+(\frac{-14}{2})^2[/tex]

Simplify

[tex]169=l^2-14l+49[/tex]

Rewrite right side a square binomial

[tex]169=(l-7)^2[/tex]

Take the square root of both sides

[tex]13=l-7\\[/tex]

Add 7 to both sides

[tex]20=l[/tex]

to solve for width simply subtract 14 from the length which is 20, so the width is 6

Width: 6

L: 20

The probability of getting a cookie with no more than 3 chips is 0.714.

It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.

We have:

Well-mixed cookie dough will produce cookies with a mean of 6 chocolate chips apiece.

Using poison ratio:

[tex]\rm P (X = k) = \dfrac{\lambda^k e^{-\lambda}}{k!}[/tex]

λ is the mean number of chocolate chips in a piece

[tex]\rm P (X = 6) = \dfrac{6^k e^{-6}}{k!}[/tex]

P(X ≥ 5) = 1 - P(X < 5)

P(X ≥ 5) = 1 - P(X = 0) + P(X = 1)  + P(X = 2)  + P(X = 3) + P(X = 4)  

[tex]\rm P(X \geq 5) = 1-[\dfrac{6^0 e^{-6}}{0!}+\dfrac{6^1 e^{-6}}{1!}+\dfrac{6^2 e^{-6}}{2!}+\dfrac{6^3 e^{-6}}{3!}+\dfrac{6^4 e^{-6}}{4!}][/tex]

After solving;

P(X ≥ 5) = 0.714

Thus, the probability of getting a cookie with no more than 3 chips is 0.714.

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The coefficient (m) of the expression represents: C. the monthly depreciation value of the gadget.

Depreciation can be defined as a process in which the monetary value of a physical asset decreases or falls over a period of time, especially due to wear and tear.

In this scenario, we can infer and logically deduce that the coefficient (m) of the given expression represents the monthly depreciation value of Megan's gadget.

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Answer:

it is C

Step-by-step explanation:

give the other guy brainliest, I am ok with hearts and good ratings.

The result of the respective questions are:

a)

This chi-square test only takes into consideration one variable.

b)

The  type of chi-square test this is, is a Goodness of Fit

To test the hypothesis, we must determine whether the actual data conform to the assumed distribution.

The "Goodness-of-Fit" test is a statistical hypothesis test that determines how well the data that was seen resembles the data that was predicted.

c)

Parameter

n = 4

Therefore

Degrees of freedom

df= n - 1

df= 4 - 1

df= 3

d)

In conclusion

Parameters

[tex]\alpha = 0.05[/tex]

df = 3

Hence

Critical value = 7.814728

Test statistic = 6.6

Test statistic < Critical value, .

NO, the result of this test is not statistically significant.

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Answer:

a. n+5

b. n+ 7

c. 1\2(n×2n)

d. n +n+1 n+2

[tex]\Large\maltese\underline{\textsf{A. What is Asked}}[/tex]

Which equation has a slope of -2 and a y-intercept of 4?

[tex]\Large\maltese\underline{\textsf{B. This problem has been solved!}}[/tex]

All of these equations are in [tex]\bf{y=Mx+b}[/tex] form.

The y-int. is b. The slope is M.

[tex]\bf{y\!=\!\!-2x+\!4}[/tex] | put in the values

[tex]\cline{1-2}[/tex]

[tex]\bf{Result:}[/tex]

                [tex]\bf{=\!\!y=-2x+4}[/tex]

[tex]\LARGE\boxed{\bf{aesthetics\not1\theta l}}[/tex]

Using the Central Limit Theorem, nothing can be stated about the shape of the sampling distribution for the sample mean, as the sample size is less than 30.

It states that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the sampling distribution is also approximately normal, as long as n is at least 30.

In this problem, we have a skewed variable and n < 30, hence nothing can be stated about the shape of the sampling distribution for the sample mean.

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Answer:

21.6 in

Step-by-step explanation:

Imagine the suitcase as a rectangle, with length of 18 in and width of 12 in. In order to find the diagonal length, simply use the Pythagorean Formula, a^2 + b^2 = c^2, and solve for c!

In this case, a = 18 and b = 12.

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The total finance charge is $611.20 (Option c) for this compound interest.

Data Given

Principal, P = $4250

Rate of Compound Interest, R = 13.25%

Time, t = 24 months, i.e., 2 years

Since it is compounded monthly, n = 12

Calculating the Total Finance Charge

We know that the formula for total amount of finance charge in a Compound Interest is,

[tex]A=P\frac{r(1+r)^{n} }{(1+r)^{n} - 1}[/tex]

Substituting the values of P, R and n, we get,

[tex]A=4250\frac{13.25(1+13.25)^{24} }{(1+13.25)^{24} - 1}[/tex]

[tex]A = 611.20[/tex]

Thus, the total finance charge is $611.20

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The net cost of the amount needed in filling the prescription is 7.86 per gallon

Net cost = 12 - (34.5% of 12)

= 12 - (0.345 × 12)

= 12 - 4.14

= 7.86 per gallon

Therefore the net cost of the amount needed in filling the prescription is 7.86 per gallon

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When t is between 2 and 5, the ball will be above 10 feet.so answer is:

C. 2 ≤ t ≤ 5

Given, height h(t) = ₋t² ₊ 7t

At what values of time t is the ball 10 feet above the ground = ?

To find out the time we need to set up an inequality.

Inequality expression is given as = ₋t² ₊ 7t ≥ 10

                                                       = ₋t² ₊ 7t ₋ 10 ≥ 0

                                                       = ₋(t² ₋ 7t ₊ 10) ≥ 0

                                                       = t² ₋ 7t ₊10 ≥ 0

Now, factorize the expression and get the factors.

= t² ₋ 2t ₋ 5t ₊ 10 ≥ 0

=t(t ₋ 2) ₋ 5(t ₋ 2) ≥ 0

=(t ₋ 2) (t ₋ 5) ≥ 0

Hence t value ranges from t = 2 and 5.

As a result, when t is between 2 and 5 seconds, the ball will be farther than 10 feet, 2 ≤ t ≤ 5

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Answer:

11:00

Step-by-step explanation:

here we have to calculate the LCM of 20 and 18

20 = 2² × 5

18 = 2 × 3²

Then

LCM(20 , 18) = 2² × 5 × 3² = 20 × 9 = 180

The

After 180 minutes (3 hours) they will have finished inspecting a car at the same point time.

Then

At 8:00 + 3 = 11:00 Ram and Robert will have finished inspecting a car at the same point time.

The value gotten when 5/8 is divided by 3/4 is 5/6.

A fraction is a non-integer that is made up of numerator and a denominator. An example is 5/8. Division is the process of grouping a number into equal parts using another number.

5/8 ÷ 3/4

5/8 × 4/3 = 5/6

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Answer:

what solution bc i could answer this pls let me know the equation but......

Step-by-step explanation:

to find a solution you can add divide or subtract to find the solution like x=2

Step-by-step explanation:

sol;

l=5w

area=l*w

or,45=5w*w

or,45=5w2

or,45\5=w2

0r[tex]\sqrt{9}[/tex]=w

or,3=w

again,

l=5w

l=5*3

l=15

Therefore, length is 15 inch and width is 3 inch.

The sum of the equation is

[tex] \frac{859}{2772} [/tex]

How to find the sum

Given the expression

s = 1 - (1)/(4) + (1)/(6)-(1)/(9)+ (1)/(11)- (1)/(14)

Let's find the LCM, which is 2772

s =

[tex] \frac{2772 - 693 + 462 - 308 +252 - 198}{2772} [/tex]

Add the numerators, do the addition before the substraction

s =

[tex] \frac{859}{2772} [/tex]

Thus, the sum of the equation is

[tex] \frac{859}{2772} [/tex]

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