There are 691 students at Palmer Elementary School and 1,293 students at Palmer Middle School. About how many students are there in all at both school?

The conversion of 68 kilometers into meters gives 68000 meters, not 0.068 meters. The error arises due to dividing instead of multiplying the kilometers by 1000 (because 1 km equals 1000m).

When you are converting from kilometers to meters, you need to understand that 1 kilometer is equal to 1000 meters. Therefore, to convert 68 kilometers to meters, you should multiply 68 by 1000, not dividing it. You calculate 68km * 1000 = 68000m. So, 68 kilometers is equal to 68,000 meters, not 0.068 meters. The response 0.068 meters is incorrect because it is vastly smaller than the actual conversion result.

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Answer:

Sample Response: With integer tiles, you would add 5 groups of negative 2 tiles or remove 2 groups of 5 positive tiles. On the number line, you would bounce by 5 to the left 2 times.

The probability that a student graduates given that the student plays one or more sports = probability that students play one or more sports and who graduate/probability that students play one or more sports

  = 0.67 / 0.72

  = 0.9306 (Answer)

The volume of the solid under the hyperbolic paraboloid [tex]z = 4 + x^2-y^2[/tex] and above the square r = [−1, 1] × [0, 2] is [tex]20\text{ cubic units}[/tex]

The question is asking us to find the volume bounded by the following surfaces

To do this, we have to evaluate the triple integral

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy\int\limit_0^{4+x^2-y^2}\,dz[/tex]

First, we integrate with respect to [tex]z[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy\int\limit_0^{4+x^2-y^2}\,dz\\\\=\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy(4+x^2-y^2)[/tex]

Next, we integrate with respect to [tex]y[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy(4+x^2-y^2)\\\\=\displaystyle \int\limit_0^2\,dx \left[4y+x^2y-\dfrac{y^3}{3} \right]_{-1}^1\\\\=\displaystyle \int\limit_0^2\,dx \left[ \left(4(1)+x^2(1)-\dfrac{(1)^3}{3} \right)- \left(4(-1)+x^2(-1)-\dfrac{(-1)^3}{3} \right)\right]\\\\=\displaystyle \int\limit_0^2\,dx \left(\dfrac{22}{3}+2x^2 \right)\\\\[/tex]

Finally, we integrate with respect to [tex]x[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \left(\dfrac{22}{3}+2x^2 \right)\\\\=\left[\dfrac{22}{3}x+\dfrac{2x^3}{3}\right]_0^2\\\\=\left(\dfrac{22}{3}(2)+\dfrac{2(2)^3}{3}\right) - \left(\dfrac{22}{3}(0)+\dfrac{2(0)^3}{3}\right)\\\\=20 \text{ cubic units}[/tex]

The volume of the solid is [tex]20\text{ cubic units}[/tex]

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Answer:

Step-by-step explanation:

answer is 15/16

The distance of the bird from its starting point will be 29.15 miles.

The Pythagoras theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

Given that;

A bird flies 25 miles Due West then turns due south and flies for another 15 miles and lands.

Now,

Since, A bird flies 25 miles Due West then turns due south and flies for another 15 miles and lands.

Let the distance of the bird from its starting point = x

So, By using Pythagoras theorem, we get;

⇒ x² = 15² + 25²

⇒ x² = 225 + 625

⇒ x² = 850

⇒ x = √850

⇒ x = 29.15 miles

Thus, The distance of the bird from its starting point = 29.15 miles.

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To find points where the tangent plane to an ellipsoid is parallel to a plane, calculate the gradient vector of the ellipsoid and set it proportional to the normal vector of the given plane, then solve the resulting system for  (x, y, z).

To find all the points where the tangent plane to an ellipsoid is parallel to a given plane, you first need to consider the equation of the ellipsoid and the equation of the tangent plane.

The ellipsoid can be described by the general equation  f(x, y, z) = 0, while the tangent plane at a point on the ellipsoid can be described by the gradient of f at that point, given as  ∇f.

The gradient, which is a vector, gives us the normal to the tangent plane at the given point.

For a tangent plane to be parallel to another plane, their normal vectors must be proportional.

So, if we have the normal vector of the given plane, we can set up an equation with the gradient of the ellipsoid, and solve for the points (x, y, z) that satisfy this condition.

It requires solving a system of equations where the coefficients of the normals to both planes are proportional.

These points  (x, y, z) will be the points of tangency where the ellipsoid's tangent plane is parallel to the given plane.

We use the calculus concepts of partial derivatives to find the gradient vector and algebra to solve for the unknowns corresponding to the points of tangency.

Mathematics uses numbers, variables, and operations, particularly in algebraic expressions. An example of this is the equation 5x + 2 = 12, where '5' and '2' are numbers, 'x' is variable, and '+' and '=' are operation symbols.

The subject that uses numbers, variables, and operations is Mathematics. This combination is commonly seen in algebraic expressions. For example, in the equation 5x + 2 = 12, '5' and '2' are numbers, 'x' is a variable, and '+' and '=' are operation symbols. The numbers are used as constants or coefficients, the variable represents an unknown value, and the operations symbols dictate how these elements should interact. Through algebra, we can solve this equation and find the value of the variable 'x'.

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In Mathematics, particularly Algebra, numbers, variables and operation symbols are frequently used. This language is evident in forms like scientific and exponential notation, which are methods to represent large quantities using powers of ten.

The system of using numbers, variables, and operation symbols is commonly seen in Mathematics. In particular, this is prominent in Algebra, where numbers are often represented by variables, and operations like multiplication, division, addition, or subtraction involve these variables. One form of this language includes scientific notation. Scientific notation is a mathematical expression used to represent very large numbers using powers of ten. For instance, 500,000,000 can be written as 5 × 10^8 in scientific notation. Another way to express large quantities is through exponential notation, where the number is represented as a product of two numbers, one of which is a power of ten. Learning and practicing how to use these forms becomes essential for further scientific studies.

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Answer:

[tex]y=3x+1[/tex]

Step-by-step explanation:

The line has positive slope, therefore the variable in the equation must be positive ([tex]y=-3x+1[/tex] and [tex]y=-3x-1[/tex] must be discarded)

Now, in the graph we can see that the line passes over the following points:

[tex](0,1) \rightarrow x=0, y=1\\(1,4) \rightarrow x=1, y=4[/tex]

With the point [tex](0,1)[/tex], we can discard [tex]y=3x-1[/tex] because:

in the equation  [tex]y=3x-1[/tex], we have: [tex]x=0 \rightarrow y=3(0)-1=0-1=-1\rightarrow y=-1[/tex]

The line doesn't pass over the point [tex](0,-1)[/tex]

Therefore, the equation is [tex]y=3x+1[/tex].

We can verify the answer with the points [tex](0,1)[/tex] and [tex](1,4)[/tex], replacing values in the equation:

[tex](0,1):\\x=0\rightarrow y=3(0)+1=0+1=1\rightarrow y=1\\\\(1,4):\\x=1\rightarrow y=3(1)+1=3+1=4\rightarrow y=4[/tex]

If the gross income is $34,100,113.67 was deducted from each paycheck for his 401(k) plan.

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.

The usage of percentages is widespread and diverse. For instance, numerous data in the media, bank interest rates, retail discounts, and inflation rates are all reported as percentages. For comprehending the financial elements of daily life, percentages are crucial.

It is given that last year, a janitorial supervisor had a gross income of $34,100 of which he contributed 8% to his 401(k) plan.

As a result,

=8% of 34,100

=0.08(34,100)

=2728

Given that there are 12 months in a year and he is paid every two months, or twice a month, for a total of 24 payments,

=2728/24

= 113.666

Thus, if the gross income is $34,100,113.67 was deducted from each paycheck for his 401(k) plan.

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Answer:

The height of the original triangle is [tex]5\ inches[/tex]

Step-by-step explanation:

we know that

The scale factor is equal to divide the length of the corresponding side of the reduced triangle by the length of the corresponding side of the original triangle

Let

z-----> the scale factor

x----->  the length of the corresponding side of the reduced triangle

y-----> the length of the corresponding side of the original triangle

so

[tex]z=\frac{x}{y}[/tex]

in this problem we have

[tex]z=0.4[/tex]

[tex]x=2\ in[/tex] -----> the height of the reduced triangle

substitute and solve for y

[tex]0.4=\frac{2}{y}[/tex]

[tex]y=2/0.4=5\ in[/tex]

When a 7-foot board is divided into 4 equal pieces, each piece will be 1.75 feet, or, as a mixed number, 1 3/4 feet.

To figure out the length of each piece when a 7-foot board is divided into 4 equal parts, we perform division. We divide the total length of the board by the number of pieces. That is, 7 ÷ 4 = 1.75.

This is a decimal number. To express it as a mixed number, we remember that '.75' is the same as '75/100', and this fraction can be simplified to '3/4'. So, 1.75 feet is the same as 1 3/4 feet. Therefore, each piece of board will be 1 3/4 feet long.

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Answer:

∠B = 19°

Step-by-step explanation:

Given : In triangle ABC, if ∠A = 120°, a = 8, and b = 3

We have to find the measure of B that is ∠B

Consider the given triangle ABC,

Using Sine rule ,

For a triangle with measure of angle A, B and C and side  a faces angle A,  

side b faces angle B and  side c faces angle C

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]

we have, a = 8 , b = 3 and ∠A = 120°

Consider first two ratios,

[tex]\frac{8}{\sin 120^{\circ}}=\frac{3}{\sin B}[/tex]

Solve for B, we have,

[tex]\sin \left(120^{\circ \:}\right)=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{8}{\frac{\sqrt{3}}{2}}=\frac{3}{\sin \left(B\right)}[/tex]

[tex]\mathrm{Apply\:fraction\:cross\:multiply:\:if\:}\frac{a}{b}=\frac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot \:c[/tex]

[tex]8\sin \left(B\right)=\frac{\sqrt{3}}{2}\cdot \:3[/tex]

Simplify, we have,

[tex]\sin \left(B\right)=\frac{3\sqrt{3}}{16}[/tex]

Taking sine inverse both side, we have,

[tex]B=\sin^{-1}\left(\frac{3\sqrt{3}}{16}\right)[/tex]

We have, [tex]B=18.95^{\circ \:}[/tex]

Thus, ∠B = 19°

The probability that at least one of the five light bulbs is defective is approximately 26.61%. This is found by calculating the probability that all bulbs work ([tex]0.94^5[/tex]) and subtracting that from 1.

The student is asking about the probability of having at least one defective light bulb in a pack of five, given a defect rate of 6%. To calculate this, we will use the complement rule, which states that the probability of at least one success is equal to 1 minus the probability of zero successes (no defective bulbs).

First, we calculate the probability that a bulb is not defective, which is 94% or 0.94 (since 100% - 6% = 94%). Since the bulbs are independent, the probability that all five bulbs are not defective is:

[tex](0.94)^5[/tex]

The probability that at least one bulb is defective is thus:

1 - [tex](0.94)^5[/tex]

Using a calculator, this gives us:

[tex]1 - (0.94^5) \approx 1 - 0.7339 \approx 0.2661 or 26.61%[/tex]

So, there is approximately a 26.61% chance that at least one of the five light bulbs is defective.

The probability that at least one of the light bulbs is defective is approximately 0.264909 or 26.49%.

To find the probability that at least one of the light bulbs is defective, we can use the complement rule.

The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.

In this case, the event of interest is that at least one light bulb is defective. The complement of this event is that none of the light bulbs are defective.

Given that 6% of the light bulbs are defective, the probability that any one light bulb is not defective is 1 - 0.06 = 0.94.

Since each light bulb operates independently of the others, the probability that none of the five light bulbs are defective is [tex]\((0.94)^5\)[/tex].

Therefore, the probability that at least one of the light bulbs is defective is [tex]\(1 - (0.94)^5\)[/tex].

Let's calculate it:

P(at least one defective) = 1 - (0.94)^5

P(at least one defective) = 1 - 0.735091 = 0.264909

So, the probability that at least one of the light bulbs is defective is approximately 0.264909 or 26.49%.

a. Yule-Simons PMF is valid due to non-negativity and sum 1 proof.

b. E[X] = 2 calculated through partial fraction and cancellation.

c. E[X^2] = ∞ shown using comparison test with p-series.


a. Proving it's a Probability Mass Function (PMF):

Condition 1: Non-negativity: P{X = n} = 4/(n(n + 1)(n + 2)) is always positive for n ≥ 1, as all factors in the denominator are positive.

Condition 2: Sums to 1:

We need to show ∑_(n=1)^∞ P{X = n} = 1.

Use partial fraction decomposition:

4/(n(n + 1)(n + 2)) = 1/(n) - 1/(n + 1) + 1/(2(n + 2))

Expand the infinite series:

∑_(n=1)^∞ P{X = n} = (1/1 - 1/2 + 1/6) + (1/2 - 1/3 + 1/8) + ...

Notice terms cancel out:

= 1 + (1/6 - 1/6) + (1/8 - 1/8) + ... = 1

Therefore, P{X = n} is a valid PMF.

b. Expectation E[X] = 2:

E[X] = ∑_(n=1)^∞ n * P{X = n}

Substitute P{X = n} with its expression:

E[X] = ∑_(n=1)^∞ n * (4/(n(n + 1)(n + 2)))

Apply partial fraction decomposition (as in a) and simplify:

E[X] = ∑_(n=1)^∞ (1/(n + 1) - 2/(n + 2))

Expand the series and observe cancellations:

E[X] = (1/2 - 2/3) + (1/3 - 2/4) + ... = 1 - 1/2 = 1/2

Multiply by 4 to account for the 4 in the original PMF:

E[X] = 4 * (1/2) = 2

Therefore, E[X] = 2.

c. Expectation E[X^2] = ∞:

E[X^2] = ∑_(n=1)^∞ n^2 * P{X = n}

Substitute P{X = n} and simplify:

E[X^2] = ∑_(n=1)^∞ (4n/(n + 1)(n + 2))

Use the comparison test:

4n/(n + 1)(n + 2) > 4n/(n^3) = 4/(n^2) for n ≥ 1

Since ∑_(n=1)^∞ 4/(n^2) (p-series with p = 2) converges, so does E[X^2].

Therefore, E[X^2] = ∞.

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