Of 300 students in the cafeteria 140 had lunch. Write the ratio of the students in the cafeteria to the students that had lunch

Use calculus to find the area of the triangle with the vertices (0, 5), (2, -2), and (5, 1).

guajiro [1.7K]

The area of the triangle with the vertices (0, 5), (2, -2), and (5, 1) by using the calculus is 21 square unit.

We need to find the equation among all possible pairs and then integrate the equations from one co-ordinate to another co-ordinate

Equation of line passing through (0,5) and (2,-2) is

y-5 = [(-2-5)/(2-0)](x-0)

=>y-5 = (-7) /2x

=>y= (-7/2x)+5 -------(eq1)

Equation of line passing through (0,5) and (5,1) is

y-5 = [(1-5) / (5-0)](x-0)

=>y-5 = (-4/5)x

=>y = (-4/5)x+5-------(eq2)

Equation of line passing from (2,-2) and (5,1) is

y-(-2) = [[1 - (-2)] / (5-2)] / (x-2)

=>y+2=(3/3)(x-2)

=>y=x-4--------(eq3)

Now, we use definite integration to find the area between the different equation of line.

So, area enclosed between the equations is given by the

area = $\int\limits^5_2$ [(-4/5)x+5 - (-7/2)x + 5)dx + $\int\limits^5_1$ [(-4/5)x+5 -(x-4)]dx

=>area= $\int\limits^5_2$ (7/2-4/5)x dx + $\int\limits^5_1$ ((-9/5)x+9)dx

Using properties of integration, $\int\limits x\, dx=x^{2}/2$

=>area= $\int\limits^5_2$ (27/10)x dx + $\int\limits^5_1$ (-9/5)x+9)dx

=>area=([27/10)×[5² - 2²])/2 + [ (-9/5)×(5²-1²) ]/2 +9×(5-1)

=>area=(27/20)×(25-4) + (-9/5)×24+9×4

=>area = (27×21)/20 + (-216)/5+ 36

=>area=(567/20) - (216/5) + 36

=>area= [(567-261×4)+(36×20)]/20

=>area=[(567-864)+720]/20

=>area=423/20

=>area=21 square unit.

Hence, area of triangle is 21 square unit.

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4 0

1 year ago

For the annual rate of change of +70%, find the corresponding growth or Decay Factor.

omeli [17]

Hello there!
1. Okay. So 70% is positive, so it is a corresponding growth. 70% is 0.7 in decimal form. C and D are both eliminated, because that's too large. When something grows by 70%, that is less than double. 0.7 is not right either, because that's more of a decay than a growth. The only answer that makes sense id 1.70, because you are going up, and adding 1 to decimal form brings you to the decimal that when multiplied will bring you to the total. The answer is B: 1.70.

2. So, -75% is negative, so it is a decay factor. A is eliminated, because it makes no sense and B is out, because that represents growth, not decay. You lose 75% of the amount overtime compounded, but you still have 25% that remains. To find that amount, you would subtract that amount from 1 to get the decay. 1 - 0.75 is 0.25. The decay factor is 0.25. The answer is D: 0.25.

4 0

3 years ago

The water works commission needs to know the mean household usage of water by the residents of a small town in gallons per day.

valina [46]

Answer:

A sample of 179 is needed.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1 - 0.85}{2} = 0.075$

Now, we have to find z in the Ztable as such z has a pvalue of $1 - \alpha$ .

That is z with a pvalue of $1 - 0.075 = 0.925$ , so Z = 1.44.

Now, find the margin of error M as such

$M = z\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

A previous study found that for an average family the variance is 1.69 gallon?

This means that $\sigma = \sqrt{1.69} = 1.3$

If they are using a 85% level of confidence, how large of a sample is required to estimate the mean usage of water?

A sample of n is needed, and n is found for M = 0.14. So

$M = z\frac{\sigma}{\sqrt{n}}$

$0.14 = 1.44\frac{1.3}{\sqrt{n}}$

$0.14\sqrt{n} = 1.44*1.3$

$\sqrt{n} = \frac{1.44*1.3}{0.14}$

$(\sqrt{n})^2 = (\frac{1.44*1.3}{0.14})^2$

$n = 178.8$

Rounding up

A sample of 179 is needed.

7 0

3 years ago

Write each trigonometric ratio as a fraction in simplest form. show all supporting work

Nesterboy [21]

Answer: $\frac{x}{20}$

Step-by-step explanation:

Sine uses the ratio, $\frac{opposite}{hypotenuse}$ so we can plug in the information we know. since there is no current value for the opposite side of angle A, we can just use a variable to replace it.

$sinA=\frac{x}{20}$

You can't reduce this any further because there is an unknown variable.

7 0

3 years ago

Evaluate 5j +12 foreach listed<br> value ofn.<br> j= 4<br> j = 6

Bezzdna [24]

Answer: 32 and 42

Step-by-step explanation:

5j +12

j= 4

j = 6

5(4) + 12

5 times 4 = 20

20 + 12 = 32

5(6) + 12

5 times 6 = 30

30 + 12 = 42

6 0

3 years ago

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