In a certain school, course grades range from 0 to 100. Adrianna took 4 courses and her average course grade was 90. Roberto took 5 courses. If both students have the same sum of course grades, what was Roberto's average?

(I want to know the most easiest way to do it)

Answer and explanation:

Use the distributive property in two different ways to find the product of 127 and 32.

Distributive property says that,

[tex]a(b+c)=ab+ac[/tex]

We have to find product of 127 and 32,

1) Split 127 as 100+27

[tex]127\times32=(100+27)\times 32[/tex]

Apply distributive property,

[tex]127\times32=100\times 32+27\times 32[/tex]

[tex]127\times32=3200+864[/tex]

[tex]127\times32=4064[/tex]

2) Split 32 as 30+2

[tex]127\times32=127\times (30+2)[/tex]

Apply distributive property,

[tex]127\times32=127\times 30+127\times 2[/tex]

[tex]127\times32=3810+254[/tex]

[tex]127\times32=4064[/tex]

Answer:

2000 points

Step-by-step explanation:

Topic: Substraction, Addition and Products

You have to organize the problem, as you can see

at Start of Round 3 David Has: - 1500

During Round 3:

5 Questions: 1000 each correctly

so 5 x 1000 = 5000

3 Questions: - 500 each incorrectly

so 3 x -500 = -1500

at the End of Round 3:

Starting: -1500

During Round 3: 5000 - 1500 = 3500

the end of round 3: -1500 + 3500 = 2000

Let

[tex]A(-4,9)\\B(11,9)\\C(5,-1) \\D(-10,-1)\\E(-4.-1)[/tex]  

using a graphing tool  

see the attached figure to better understand the problem

we know that

Parallelogram is a quadrilateral with opposite sides parallel and equal in length

so

[tex]AB=CD \\AD=BC[/tex]

The area of a parallelogram is equal to

[tex]A=B*h[/tex]  

where  

B is the base

h is the height  

the base B is equal to the distance AB

the height h is equal to the distance AE  

Step 1

Find the distance AB

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

[tex]A(-4,9)\\B(11,9)[/tex]  

substitute the values

[tex]d=\sqrt{(9-9)^{2}+(11+4)^{2}}[/tex]

[tex]d=\sqrt{(0)^{2}+(15)^{2}}[/tex]

[tex]dAB=15\ units[/tex]

Step 2

Find the distance AE

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

[tex]A(-4,9)\\E(-4.-1)[/tex]  

substitute the values

[tex]d=\sqrt{(-1-9)^{2}+(-4+4)^{2}}[/tex]

[tex]d=\sqrt{(-10)^{2}+(0)^{2}}[/tex]

[tex]dAE=10\ units[/tex]

Step 3

Find the area of the parallelogram

The area of a parallelogram is equal to

[tex]A=B*h[/tex]

[tex]A=AB*AE[/tex]

substitute the values

[tex]A=15*10=150\ units^{2}[/tex]

therefore

the answer is

the area of the parallelogram is [tex]150\ units^{2}[/tex]

The area of the parallelogram is 0 units.

To find the area of a parallelogram, we can use the formula A = base * height. We can find the length of the base by finding the distance between points A and D, which is 11 units. To find the height, we can find the distance between points A and B, which is 0 units. Therefore, the area of the parallelogram is 0 units.

Answer:16

Step-by-step explanation:

Answer:

(C) is one more than the power

Step-by-step explanation:

Let us consider a binomial expansion:

[tex](x+y)^2=x^2+2xy+y^2[/tex]

The power of the above expansion is [tex]2[/tex] and the number of the terms in the above binomial expansion are [tex]3[/tex], thus the total number of terms in the binomial expansion is one more than the power.

Hence, Option C is correct.

Explanation on finding distance from a point to the origin as a function of x and showing invariance under rotations.

Distance from point P to the origin as a function of x:

Given point P(x, y) on y = x² – 9.

The distance d from P to the origin is d = √(x² + y²).

Substitute y = x² - 9 into the distance formula to express d as a function of x: d = √(x² + (x² - 9)²).

Invariance of distance under rotations:

The distance from P to the origin remains constant regardless of the rotation, as it depends only on the coordinates of the point and not the orientation of the coordinate system.

The slices of watermelon each person will get is  1.5 slices.

If the watermelon is divided equally, the number of slices each person would get can be determined by dividing the total slices of watermelon by the total number of friends.

Slice each person would get = total slices / total number of friends

6 / 4 = 1.5 slices

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The Java Rectangle class implements both the Shape and Serializable interfaces. Therefore, Rectangle r can be legally assigned to Rectangle, Shape, Serializable, and Object data types. It can't be assigned to a String, Measurable, and ActionListener as Rectangle doesn't implement these.

In Java, the Rectangle class implements the Shape and Serializable interfaces. Therefore, the legal assignments would be:

The rest assignments are not legal because the Rectangle class doesn't implement Measurable, ActionListener interfaces and it cannot be assigned to a String.

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Answer:

Option C is correct.

Step-by-step explanation:

The formula is = [tex](1+\frac{r}{n})^{n}-1[/tex]

r = rate of interest

n = number of times its compounded

1. 4.0784% compounded monthly

here n = 12

[tex](1+\frac{0.040784}{12})^{12} -1[/tex] = 1.0403-1 = 0.0403

2. 4.0798% compounded semiannually

here n = 2

[tex](1+\frac{0.040798}{12})^{2} -1[/tex] = 1.0066-1 = 0.0066

3. 4.0730% compounded daily

here n = 365

[tex](1+\frac{0.040730}{12})^{365}[/tex] = 3.328-1 = 2.328

65*8 = 520 miles traveled

520/80 = 6.5 hours for the car

Answer:

The automobile would take 6.5 hours to complete the trip.

Step-by-step explanation:

This is a case of a inverse variation relationship because when you increase the speed of traveling the time of arriving will reduce, now we have to define the equation and the variables:

Inverse variation equation: [tex]x1y1=x2y2[/tex]

[tex]x1[/tex]: speed of the bus, 65 miles per hour

[tex]y1[/tex]: time to Austin by bus, 8 hours

[tex]x2[/tex]: speed of the automobile, 80 miles per hour

[tex]y2[/tex]: time to Austin by automobile, unknown

Now we can replace the values in the equation and clear [tex]y2[/tex], this is:

[tex]65*8=80*y2\\y2=\frac{65*8}{80} =\frac{520}{80} =6.5\\[/tex]

The automobile would take 6.5 hours to complete the trip.

The annual value of the employer's contribution towards Trina's health insurance plan is $5,580.

Here, we have to find the annual value of the employer's contribution, we need to determine how much the employer pays towards the health insurance plan in one month.

Trina's employer purchased a health insurance plan that costs $550 per month, and Trina pays $85 toward the plan each month.

Therefore, the employer's contribution is the difference between the total cost of the plan and what Trina pays:

Employer's contribution = Total cost of the plan - Trina's contribution

Employer's contribution = $550 - $85

Employer's contribution = $465 per month

Now, to find the annual value of the employer's contribution, we multiply the monthly contribution by 12 (since there are 12 months in a year):

Annual employer's contribution = $465 * 12

Annual employer's contribution = $5,580

So, the annual value of the employer's contribution towards Trina's health insurance plan is $5,580.

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The probability that both alternators fail is approximately 0.0004, the probability that neither fails is approximately 0.9612, and the probability that at least one fails is approximately 0.0388, all rounded to four decimal places.

(a) The probability that both Alternator will fail is calculated by multiplying the probabilities of each failing:
P(Both Fail) = P(Alternator 1 Fails) x P(Alternator 2 Fails) = 0.019 x 0.019 ≈ 0.000361 = 0.0004 (rounded to four decimal places).

(b) The probability that an alternator does not fail is 1 minus the probability that it fails.
P(Neither Fail) = (1 - P(Alternator Fails))^2 = (1 - 0.019)^2 ≈ 0.9612 (rounded to four decimal places).

(c) To find the probability of at least one alternator failing, we subtract the probability of neither failing from 1:
P(At Least One Fails) = 1 - P(Neither Fail) = 1 - 0.9612 ≈ 0.0388 (rounded to four decimal places)

The speed of the baseball when it passes over the home plate is 97.67 mph.

Given :

Differentiate the function x(t) and y(t) with respect to t.

[tex]x'(t)=143[/tex]

[tex]y'(t)= -32t+5[/tex]   ---- (3)

Now, put the value of x(t) in equation (1).

60.5 = 143t

t = 0.423 sec

Now, put the value of t in equation (3).

[tex]y'(t) = -32(0.423)+5=-8.536[/tex]

Now, [tex]s(t) = \sqrt{(x'(t))^2+(y'(t))^2}[/tex]

[tex]s(t)=\sqrt{143^2+(-8.536)^2}[/tex]

[tex]\rm s(t)=\sqrt{20521.8633} = 143.2545\;ft/sec[/tex]

[tex]\rm s(t) = 97.67 mph[/tex]

Therefore, the speed of the baseball when it passes over the home plate is 97.67 mph.

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The area of the shaded figure is (x⁴ + 19x² + 100) square meters after subtracting the area of the smaller square from the area of the larger square.

It is defined as a two-dimensional geometry that has four sides and four vertices. The sides of the square are equal in length. It is a regular quadrilateral.

It is given that:

Large square side length: (x squared plus 10)

Small square side length: x

The area of the large square = (x² + 10)(x² + 10)

The area of the large square = (x² + 10)²

The area of the small square = (x)(x)

The area of the small square = x²

The area of the shaded figure = (x² + 10)² - x²

The area of the shaded figure = (x⁴ + 19x² + 100) square meters

Thus, the area of the shaded figure is (x⁴ + 19x² + 100) square meters after subtracting the area of the smaller square from the area of the larger square.

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To prove that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex] and [tex]o(x^3)[/tex], we must show how the function grows in comparison to x^3 asymptotically.

To prove that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex] and [tex]o(x^3)[/tex], we need to show two things:

1. Proving that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex]

For a function to be [tex]\omega(x^3)[/tex], it needs to grow faster than [tex]x^3[/tex] asymptotically. This requires that the ratio of the function to [tex]x^3[/tex] tends towards infinity as x approaches infinity.

Consider:

[tex]\lim_{x \to \infty}[\frac{f(x)}{x^3} ] = \lim_{x \to \infty}[ \frac{x^3-1000x^2+x-1}{x^3} ][/tex]

Simplify the expression:

[tex]\lim_{x \to \infty}[ 1-\frac{1000}{x}+\frac{1}{x^2}-\frac{1}{x^3} ] =1[/tex]

Since the limit does not tend to infinity but instead tends to 1, f(x) is not ω(x^3). We made a mistake in our earlier assumption; let's correct this in the next point.

2. Proving that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]o(x^3)[/tex]

To show that f(x) is [tex]o(x^3)[/tex], the ratio of f(x) to [tex]x^3[/tex] should tend to zero as x tends to infinity.

Consider:

[tex]\lim_{x \to \infty} [\frac{f(x)}{x^3}] = \lim_{x \to \infty}[\frac{x^3-1000x^2+x-1}{x^3}][/tex]

Simplify the expression:

[tex]\lim_{x \to \infty}[ 1-\frac{1000}{x}+\frac{1}{x^2}-\frac{1}{x^3} ] =1[/tex]

This indicates that the limit tends to 1 and not 0, hence f(x) is not [tex]o(x^3)[/tex] either. We can see in both cases that the limits did not meet the required criteria for [tex]\omega(x^3)[/tex] or [tex]o(x^3)[/tex], implying a misunderstanding in the problem setup.

Final answer:

Using the formula for the confidence interval of a proportion, and after calculating standard error and margin of error, the 98% confidence interval for the proportion of 8th graders involved in after school activities is [0.799, 0.981]. Therefore, option a is correct.

Explanation:

To find the 98% confidence interval for the proportion of 8th graders involved in some type of after school activity, we can use the formula for the confidence interval of a proportion:

CI = ± z * √((p*(1-p))/n), where p is the sample proportion, n is the sample size, and z is the z-score corresponding to the confidence level.

In this case, we have p = 0.89 (since 89% of the sample is involved in after school activities), n = 64, and for a 98% confidence interval, the z-score (from z-tables or statistical software) is approximately 2.33.

First, let's calculate the standard error (SE): SE = √((0.89*(1-0.89))/64) = √((0.89*0.11)/64) = √(0.0989/64) = 0.0392.

Next, calculate the margin of error (ME): ME = z * SE = 2.33 * 0.0392 = 0.09136.

Now we can calculate the confidence interval: the lower limit is p - ME = 0.89 - 0.09136 = 0.79864 and the upper limit is p + ME = 0.89 + 0.09136 = 0.98136.

After rounding to three decimal places, the 98% confidence interval for the proportion is [0.799, 0.981]. Hence, option a is correct.