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Anuta_ua [19.1K]
3 years ago
5

Determine if the lines r1(t) = <3, 0, 2> + t <1, 2, −2> and r2(s) = <0, 1, −1> + s <4, 1, 1> intersect,

and if they do, find the point of intersection.
Mathematics
1 answer:
Temka [501]3 years ago
4 0

Answer:

Yes lines are intersecting, point of intersection is <4,2,0>.

Step-by-step explanation:

Given parametric equations of line are:

r_{1}(t) =  + t  \\ r_{1}(t) = \\ r_{1}(t) = ---(1)

r_{2}(s) =  + s \\ r_{2}(s) = \\ r_{2}(s) = ---(2)\\

If lines are intersecting then parametric coordinates of (1) are equal to (2)

3+t=4s---(A)\\2t=1+s---(B)\\2-2t=s-1---(C)\\

Considering A and B to find values of t and s

From A

t=4s-3---(D)

Putting in (B)

2(4s-3)=1+s

8s-6=1+s

7s=7

s=1

Then

t=4-3

t=1

If lines are intersecting then these values of s and t must satisfy (C)

2-2(1)=1-1

0=0

This shows lines are intersecting.

At this value of t, (1) becomes

r_{1}(1) = \\=

Putting s=1 in (2)

r_{2}(1)=4, 1+1,-1+1>\\=

Point of intersection is <4,2,0>.

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Point C is the Image of C (-4,-2) under a Reflection across the x-axis.
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Step-by-step explanation:

\text {Reflection Rule for across the X-axis: } (x,y) \rightarrow (x,-y)

Using the point C (-4,-2) and the refection rule:

(-4,-2) \rightarrow(-4, 2)

\text {C' should be (-4,2)}

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Peter borrowed a sum of R12 850.00 at the beginning of a year with a certain simple interest rate. After 14 quarters, in a diffe
ladessa [460]

The original annual simple interest rate, rounded to two decimal places, is 3.79%

What is the formula for simple interest?

The simple interest on a loan or deposit is determined as the principal multiplied by the simple interest rate and time

I=PRT

The first loan:

P=12 850.00

R=r(assume it is r)

T=4 years

I=12 850.00*r*4

I=51400r

The second loan was taken after 14 quarters the first was taken out, which is the same as after 3.5 years, hence, the interest on the second loan is only for a half a year

P=3 273.00

R=0.5r( half of the interest on the first loan)

T=0.5 years

I=3 273.00*0.5r*0.5

I= 818.25r

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Total interest=52218.25r

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1 980.00=52218.25r

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r=3.79%

Find out more about simple interest on: brainly.com/question/1115815

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2 years ago
Julia knows that 65×4,903=318,695. Use this understanding to help Julia solve 6.5×4903
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Answer:

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Step-by-step explanation:

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Consider the computer output below. Fill in the missing information. Round your answers to two decimal places (e.g. 98.76). Test
slamgirl [31]

Answer:

SE_{Mean}=\frac{s}{\sqrt{n}}=\frac{4.77}{\sqrt{19}}=1.094

t=\frac{98.77-100}{\frac{4.77}{\sqrt{19}}}=-1.124      

The 95% confidence interval would be given by (96.625;100.915)  

a) df=n-1= 19-1= 18

b) p_v =2*P(t_{18}      

If we compare the p value and a significance level for example \alpha=0.05 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.

c) The only thing that changes is the p value and would be given by:

p_v =P(t_{18}>-1.124)=0.862      

But again since the p value is higher than the significance level we fail to reject the null hypothesis.

Step-by-step explanation:

Previous concepts and data given

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

\bar X=98.77 represent the sample mean    

s=4.77 represent the sample standard deviation  

n=19 represent the sample selected  

\alpha significance level    

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if we have significant difference on the mean of 100, the system of hypothesis would be:    

Null hypothesis:\mu = 100    

Alternative hypothesis:\mu \neq 100    

From the central limit theorem we know that the distribution for the sample mean \bar X is given by:

\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

And we can calculate the Standard error for the mean like this:

SE_{Mean}=\frac{s}{\sqrt{n}}=\frac{4.77}{\sqrt{19}}=1.094

The confidence interval for the mean is given by the following formula:  

\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}} (1)  

Since the Confidence is 0.95 or 95%, the value of \alpha=0.05 and \alpha/2 =0.025, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that z_{\alpha/2}=1.96  

Now we have everything in order to replace into formula (1):  

98.77-1.96\frac{4.77}{\sqrt{19}}=96.625  

98.77+1.96\frac{4.77}{\sqrt{19}}=100.915  

So on this case the 95% confidence interval would be given by (96.625;100.915)  

Part a

The degree of freedom are given by:

df=n-1= 19-1= 18

Part b

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}  (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

We can replace in formula (1) the info given like this:    

t=\frac{98.77-100}{\frac{4.77}{\sqrt{19}}}=-1.124      

Then since is a two sided test the p value would be:    

p_v =2*P(t_{18}      

If we compare the p value and a significance level for example \alpha=0.05 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.

Part c

If the system of hypothesis on this case are:

Null hypothesis:\mu = 100    

Alternative hypothesis:\mu > 100  

The only thing that changes is the p value and would be given by:

p_v =P(t_{18}>-1.124)=0.862      

But again since the p value is higher than the significance level we fail to reject the null hypothesis.

7 0
3 years ago
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