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2 years ago

1/5x-2/3y=30 when y=15

Mathematics

1 answer:

dolphi86 [110]2 years ago

5 0

Answer:

Step-by-step explanation:

Step 1: Set y to 15 and solve

$\frac{1}{5}x - \frac{2}{3}*15=30$

$\frac{1}{5}x-10=30$

Add 10 to both sides

$\frac{1}{5}x-10+10=30+10$

$\frac{1}{5}x=40$

Multiply both sides by 5

$\frac{1}{5}x*\frac{5}{1}=40*\frac{5}{1}$

$x=200$

Answer: $x=200$

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The mean calculated for this case is $\bar X=584$

And the 95% confidence interval is given by:

$584-2.776\frac{86.776}{\sqrt{5}}=476.271$

$584+2.776\frac{86.776}{\sqrt{5}}=691.729$

So on this case the 95% confidence interval would be given by (476.271;691.729)

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

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

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$\bar X$ represent the sample mean for the sample

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

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

In order to calculate the mean and the sample deviation we can use the following formulas:

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)

$s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}$ (3)

The mean calculated for this case is $\bar X=584$

The sample deviation calculated $s=86.776$

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=5-1=4$

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 tabel to find the critical value. The excel command would be: "=-T.INV(0.025,4)".And we see that $t_{\alpha/2}=2.776$

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

$584-2.776\frac{86.776}{\sqrt{5}}=476.271$

$584+2.776\frac{86.776}{\sqrt{5}}=691.729$

So on this case the 95% confidence interval would be given by (476.271;691.729)

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A 12-meter ladder leans against a building forming a 30° angle with the building.

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Answer:

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A diagram is needed to see what is going on....

Without loss of generality (WLOG)

The wall is on the right. The ladder leans against the wall

with a POSITIVE slope, from SW to NE (quadrant 3 to quadrant 1).

The measure from the bottom of the ladder to the wall is 6.

Option 1:

The ladder, ground and wall form a right triangle.

The hypotenuse (ladder) is 14 feet.

The bottom of the ladder is 6 feet from the wall,

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The pythagorean theorem applies, which says

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h = sqrt(160)

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= sqrt(16)* sqrt(10)

= 4*sqrt(10) <--- exact answer

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= 12.64911....

12.65 <--- rounded to 2 digits as directed

----------------------------------------------

Option #2: using trig

With respect to the angle formed by the bottom of the

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cos T = 6/14 = 3/7

T = inverse-cosine(3/7) = 64.623006647 degrees

sin(64.623006647) = h/14

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hope this helps

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