In Exercises 11-14, find a point P where the line in- tersects the given plane or decide that the line does not intersect the plane. x =4+2t y=7+2t, z=5+41 x+y-z=2

By applying the principle of inclusion-exclusion, we can show that for any events E1, E2,..., EM, the inequality P(E1 or E2 or ... or EM) < P(EM) holds. This result holds true for any integer M ≥ 1.

To prove the statement by induction, we will assume that for M = 1, the inequality holds true. Then we will show that if the statement holds for M, it also holds for M + 1.

Base case (M = 1):

For M = 1, we have P(E1) ≤ P(E1), which is true.

Inductive step:

Assuming that the inequality holds for M, we need to show that it holds for M + 1. That is, we need to prove P(E1 or E2 or ... or EM or EM+1) < P(EM+1).

Using the principle of inclusion-exclusion, we can express the probability of the union of events as follows: P(E1 or E2 or ... or EM or EM+1) = P(E1 or E2 or ... or EM) + P(EM+1) - P((E1 or E2 or ... or EM) and EM+1). Since events E1, E2, ..., EM, and EM+1 are mutually exclusive, the last term on the right-hand side becomes zero: P(E1 or E2 or ... or EM or EM+1) = P(E1 or E2 or ... or EM) + P(EM+1)

Since we assumed that P(E1 or E2 or ... or EM) < P(EM), we can rewrite the inequality as: P(E1 or E2 or ... or EM or EM+1) < P(EM) + P(EM+1)

Now we need to show that P(EM) + P(EM+1) < P(EM+1) for the inequality to hold. Simplifying the expression, we have: P(EM) + P(EM+1) < P(EM+1)

Since P(EM+1) is a probability and is always non-negative, this inequality holds true. Therefore, by the principle of mathematical induction, we have shown that for any integer M ≥ 1, the inequality P(E1 or E2 or ... or EM) < P(EM) holds.

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The surface area of the composite solid is 286 square feet.

To calculate the surface area of this composite solid,

Find the areas of each individual shape and then add them up.

The rectangular prism has a surface area of,

2(11x2 + 11x2 + 2x11)

= 2(22 + 22 + 22)

= 2(66)

= 132 square feet.

The square pyramid has a base area of,

11x11 = 121 square feet.

The area of each triangular side can be found using the formula,

1/2 x base x height.

The base of each triangle is 11 feet (since it is the same as the length of the base of the pyramid), and the height of each triangle is 7 feet.
So each triangle has an area of 1/2 x 11 x 7 = 38.5 square feet.

There are four triangular sides, so the total area of the triangular sides is 4 x 38.5 = 154 square feet.

Adding the surface area of the rectangular prism and the square pyramid, we get

132 + 154 = 286 square feet.

Therefore, the correct answer is 286 square feet.

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Using the theories of Application of derivative,we got that  59.11221inches³/sec  is rate of volume changing when the radius is 2.3 inches.

We are given rate of change of radius (dr/dt)=0.6inches

We know very well that volume of sphere is given by (4/3)πr³

where r is the radius of the sphere.

Now, differentiating the volume with respect to radius, we get

=>dV/dr=4πr²

=>dV/dt=(dv/dr)·(dr/dt)

=>dV/dt=4πr².(dr/dt)

We are given that dr/dt=0.6inches/sec

So, dV/dt=4πr²·(0.6)

Now, putting values of r=2.3

 dV/dt=4π(2.3)²×(0.6)

=>dV/dt=4×3.14×2.3×2.3×0.6

=>dV/dt=59.11221inches³/sec

Hence, if we suppose suppose that the radius is expanding at a rate of 0.6 inches per second, the volume is changing at the rate of 59.11221inches³/sec when the radius is 2.3 inches.

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The total arc length AB that the pendulum travels is approximately 13.96 inches.

To find the total arc length traveled by the pendulum, we can use the formula for the arc length of a pendulum given by:

Arc length (s) = θ × r

where θ is the angle in radians and r is the length of the pendulum.

However, in this case, we are given the angle in degrees, so we need to convert it to radians. The formula for converting degrees to radians is:

θ (in radians) = (π/180) × θ (in degrees)

Given that the angle is 80°, we can convert it to radians as follows:

θ (in radians) = (π/180) × 80° = (4π/9) radians

Now, we can calculate the arc length:

Arc length (s) = (4π/9) radians × 10 inches

The total arc length traveled by the pendulum is therefore:

Arc length (s) = (40π/9) inches

To approximate the answer, we can use the value of π as approximately 3.14:

Arc length (s) ≈ (40 × 3.14/9) inches

Arc length (s) ≈ 13.96 inches

Therefore, the total arc length AB that the pendulum travels is approximately 13.96 inches.

Please note that this calculation assumes that the pendulum swings in a perfect arc without any friction or external factors affecting its motion. In reality, there may be slight variations due to factors such as air resistance or imperfections in the pendulum's motion.

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a) The number of intersections between the railroad and the highway is of: One.

b) The number of intersections between the railroad and the turnpike is of: Two.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which the parameters are listed as follows:

The points for the railroad are given as follows:

(-8, 0) and (0,5).

When x = 0, y = 5, hence the intercept is of:

b = 5.

When x increases by 8, y increases by 5, hence the slope is of:

m = 5/8.

Thus the equation for the railroad is of:

y = 5/8x + 5.

Then the number of intersections with the highway is of one, as they are linear functions with different slopes.

With the turnpike, the intersection is given as follows:

5/8x + 5 = x²/3

x² -1.875x - 15 = 0.

The discriminant is of:

(-1.875)² - 4(1)(-15) = 63.5.

Which is positive, hence they have two intersections.

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The answer to your question is 38.6

The appropriate simulation applet to calculate an approximate p-value is 0.0077.

The strong evidence that a short delay hinders the memorization process is strong data suggests that a brief delay impairs memorization.

Delay or no delay, answer: the number of words remembered

The mean distance between words remembered with and without delay over time is null, or zero.

Alt: The mean distance in words remembered over the long term (with no delay minus with delay) is higher than 0.

Yes, the majority of the lines do skew to the right. The mean for no delay is 10.55 whereas that for with delay is 8.7.

The mean distance in words remembered is 1.85.

1)0.0077 as the p-value

2)Strong data suggests that a brief delay impairs memorization.

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If he got -4 = 4, the equation has no solution, due that the two numbers are not equal, so they're not an equation

Answer:

the needed amount of what? community service?If so about 6 days of work on saturday and 3 school days. :)

The graph of 3x-2y≤6 is the third graph, for 3x-2y<6 is the first graph, for 3x-2y>6 is the fourth graph and for 3x-2y≥6 is the second graph. The solution has been obtained using concept of linear inequality.

What is linear inequality?

A linear inequality is one that would produce a linear equation if the equals relation were used instead of the inequality. When multiplying or dividing both sides by a negative number in order to solve the inequality, the direction of the inequality is reversed. The entire set of solutions to an inequality is known as the solution set.

We are given for graphs, of which two graphs are dotted and two are simple straight line graphs.

The dotted graphs are drawn for the inequalities having < or >

Whereas the simple straight line graphs are drawn for the inequalities having ≤ or ≥.

Now, to notice the shaded pattern, we will see whether the equations are true for (0,0) or not

1. 3x-2y≤6

⇒ 0≤6

So, the equation is true for the point.

Hence, the third graph represents this equation.

2. 3x-2y<6

⇒ 0<6

So, the equation is true for the point.

Hence, the first graph represents this equation.

3. 3x-2y>6

⇒ 0>6

So, the equation is false for the point.

Hence, the fourth graph represents this equation.

4. 3x-2y≥6

⇒ 0≥6

So, the equation is false for the point.

Hence, the second graph represents this equation.

Hence, the graphs are matched with the inequalities.

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Since, there are multiple questions so, the question answered above is attached below.

Answer:

-3=x

Step-by-step explanation:

3(x+4)-5=2x+4

use distributive property to get:

3x+12-5=2x+4

simplify to get:

3x+7=2x+4

subtract 4 from both sides to get:

3x+3=2x

subtract 3x from both sides to  get:

3=-x

multiply both sides by -1 to get your answer:

-3=x

Answer:

The answer is 13 1/8ft3

Explanation:

Substitute into the formula:

Volume = 3 1/2 x 2 1/2 x 1 1/2

= 7/2 x 5/2 x 3/2

= 105/8

= 13 1/8 ft3

Answer: the answer is D

Step-by-step explanation: BECAUSE IT IS!

Answer:

dx/dt = 5√5/2

Step-by-step explanation:

Given:

dx/dt = 25 feet per second

According to Pythagoras theorem:

x² + 90² = y²

We know that x = 30 ft

So,

x² + 90² = y²

30² + 90² = y²

y = 30√10

So,

x² + 90² = y²

By taking differentiate:

2x dx/dt = 2y dy/dt

[x/y][dx/dt] = dy/dt

dx/dt = 5√5/2

Answer:

A person bikes 1 1/2 miles at a consistent rate in 1/8 of an hour

Calculate how long the toy travels.

Step-by-step explanation: idc

Answer:

or, x2-(2-15)x +30= 0

or, x2-2x-15x + 30= 0

or,x(x-2)-15(x-2)=0

or, (x-2) (x-15)=0

either, or,

x-2=0 x-15=0

Answer:

x2-(10+3)x+30=0

x2-10x-3x+30=0

x(x-10)-3(x-10)=0

(x-10)(x-3)=0

either, or,

x-10=0 x-3=0

x=10 x=3

therefore,x=10,3

Answer:

\( \angle C \cong \angle M \)

\( \angle G \cong \angle P \)

\( \angle I \cong \angle R \)

\( \overline{CG} \cong \overline{MP} \)

\( \overline{GI} \cong \overline{PR} \)

\( \overline{CI} \cong \overline{MR} \)

Step-by-step explanation:

Given the congruence statement ∆CGI \( \cong \) ∆MPR, it follows that the corresponding sides of both ∆s are equal, as well as the corresponding vertices or angles. It implies that ∆CGI and ∆MPR are of the same shape and size.

✅Thus, the 6 pairs of the corresponding congruent parts of ∆CGI and ∆MPR are:

\( \angle C \cong \angle M \)

\( \angle G \cong \angle P \)

\( \angle I \cong \angle R \)

\( \overline{CG} \cong \overline{MP} \)

\( \overline{GI} \cong \overline{PR} \)

\( \overline{CI} \cong \overline{MR} \)

Answer:

Part i would be 2/6 or 1/3 (when simplified)

Part ii would be 4/6, or 2/3 (when simplified)

Step-by-step explanation:

There are a total of 6 possibilities you can get on a six sided dice. So for part i, your chances of getting a 1 or a 2 would be 2/6 (2 numbers you want out of 6 choices).

For Part ii, it's 4/6, because there's 4 numbers you have (3,4,5,6) out of 6 choices, that are NOT 1 or 2.

Both answers in Part A can be simplified to a simpler term. The answers in Part A, we can assume, are Equally likely to happen.

Hope this helps!

Answer:

B

Step-by-step explanation:

\(x = \frac{2}{3}\pi r^{3}\)

The objective is to isolate r and make r the subject of the equation:

Cross-multiplication is applied:

\(= 2\pi r^{3}= 3x\)

The variables and number multiplied with \(r^{3}\) are transferred to the other side of the equation:

\(= \pi r^{3} = \frac{3x}{2}\)

\(= r^{3} = \frac{3x}{2\pi}\)

Cube root is applied to both sides of the equation to get rid of the cube:

\(= \sqrt[3]{r^{3}} = \sqrt[3]{\frac{3x}{2\pi}}\)

\(r = \sqrt[3]{\frac{3x}{2\pi}}\)

Answer:

2 2/5 (For the word problem)

For the graph (In order)

4/5

1 1/4

12/30 (Or 2/5)

2 6/12 (or 2 1/2)

Step-by-step explanation:

1:

12 pounds make 5 pies, turn it into a fraction.

12/5

Now turn it into a mixed number. How many fives can go into 12?

Two fives equal ten, subtract it from twelve. Bring the amount of fives to the left.

2 2/5

2: Oh, that's really easy.

Four divided by five is 4/5

Five divided by four is 5/4 which as a mixed number is 1 1/4

3: 12 divided by thirty is 12/30, you can simplify it by dividing both sides of the denominator by three, then two. But if it's not required then just don't 12/30

4: 30 divided by 12 is 30/12, how many twelves can go into 30? Two.

30 - 24 = 6

2 6/12

Answer:

He will have it on the 15th day

a) The probability histogram of pmf for the number of students who show up for a professor's office hour on a particular day is shown below.

b) The probability that at least two students show up = 0.55 and the probability that more than two students show up = 0.25

c) The probability that between one and three students show up = 0.7

d) The probability that the professor shows up = 0.20

First we write the number of students who show up for a professor's office hour on a particular day and their pmf in tabular form.

x        p(x)

0       0.20

1       0.25

2       0.30

3       0.15

4       0.10

The probability histogram of this data is shown below.

The probability that at least two students show up would be,

P(x ≥ 2) = p(2) + p(3) + p(4)

P(x ≥ 2) = 0.30 + 0.15 + 0.10

P(x ≥ 2) = 0.55

Now the probbability that more than two students show up:

P(x > 2) = p(3) + p(4)

P(x > 2) = 0.15 + 0.10

P(x > 2) = 0.25

The probability that between one and three students show up would be:

P(1 ≤ x ≤ 3) = p(1) + p(2) + p(3)

P(1 ≤ x ≤ 3) = 0.25 + 0.30 + 0.15

P(1 ≤ x ≤ 3) = 0.7

And the probability that the professor shows up would be: p = 0.20

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To find the partial derivative of z with respect to x, we need to differentiate the equation f(x,y) with respect to x while treating y as a constant. Using the formal definition of the partial derivative, we have:

∂z/∂x = lim(h→0) [f(x+h,y) - f(x,y)]/h
= lim(h→0) [(12(x+h)² - 9(x+h)y + 16y²) - (12x² - 9xy + 16y²)]/h
= lim(h→0) [24xh + 12h² - 9yh]/h
= lim(h→0) (24x + 12h - 9y)
= 24x - 9y

Similarly, to find the partial derivative of z with respect to y, we differentiate the equation with respect to y while treating x as a constant:

∂z/∂y = lim(k→0) [f(x,y+k) - f(x,y)]/k
= lim(k→0) [12x² - 9x(y+k) + 16(y+k)² - (12x² - 9xy + 16y²)]/k
= lim(k→0) (32k - 9x)
= -9x

Therefore, the partial derivatives of z with respect to x and y are:

∂z/∂x = 24x - 9y
∂z/∂y = -9x

This is the explanation for finding the partial derivatives of z with respect to x and y using the formal definition of the partial derivative. The derivative is a measure of the rate at which a function changes with respect to its input variables, while an equation is a statement that two expressions are equal. In this case, we used an equation to represent a function of two variables and found its partial derivatives with respect to each variable using the formal definition of the partial derivative.

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Rounding to two decimal places, the upper and lower limits of the confidence interval are: Upper limit = 2,130.78 square feet, Lower limit = 1,631.22 square feet

To find the 90% confidence interval for the mean size of all houses in this city, we need to use the formula:

CI = X ± (Zα/2) * (σ/√n)

Where X is the sample mean (1,881.0 square feet), σ is the population standard deviation (328.00 square feet), n is the sample size (8), and Zα/2 is the critical value for the 90% confidence level (1.645).

Plugging in the values, we get:

CI = 1,881.0 ± (1.645) * (328.00/√8)

Simplifying the equation, we get:

CI = 1,881.0 ± 249.78

Rounding to two decimal places, the upper and lower limits of the confidence interval are:

Upper limit = 2,130.78 square feet
Lower limit = 1,631.22 square feet

Therefore, we can be 90% confident that the mean size of all houses in this city is between 1,631.22 and 2,130.78 square feet.

To calculate the 90% confidence interval for the mean size of all houses in this city, we need to use the given information:

Sample size (n) = 8
Sample mean (x) = 1,881.0 square feet
Sample standard deviation (s) = 328.00 square feet

We also need the t-distribution critical value for a 90% confidence interval and 7 degrees of freedom (n-1 = 8-1 = 7). Using a t-table or calculator, the t-value is approximately 1.895.

Next, calculate the standard error:
Standard Error (SE) = s / √n = 328 / √8 ≈ 115.99

Now, calculate the margin of error:
Margin of Error (ME) = t-value * SE = 1.895 * 115.99 ≈ 219.84

Finally, calculate the lower and upper limits of the 90% confidence interval:
Lower Limit = x - ME = 1881 - 219.84 ≈ 1661.16
Upper Limit = x + ME = 1881 + 219.84 ≈ 2100.84

So, the 90% confidence interval for the mean size of all houses in this city, rounded to two decimal places, is (1661.16, 2100.84) square feet.

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Outliers in regression can be identified by looking for data points that are far away from the overall trend of the data. These points are usually evaluated by calculating the distance from the regression line to the point and comparing it to the standard deviation of the model.

Outliers in regression can be identified by looking for data points that are far away from the overall trend of the data. These points will often have a much larger or smaller value than the rest of the data, so they can be easily spotted by eye. However, to make sure that a point is truly an outlier, it is important to calculate the distance from the regression line to the point and compare it to the standard deviation of the model. If the distance is greater than two or three times the standard deviation, then the point is likely an outlier. Outliers can have a significant effect on the outcome of a regression analysis, so it is important to identify and address them appropriately.

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Step-by-step explanation:

IF  x-4   is a factor, then putting in x = + 4    will make the polynomial = 0

5 ( 4^3) - 20 (4^2) - 5(4)  + 20   =   0    Yes...it is a factor

\(x - 4 = 0 \\ x = 4 \\ \)

substitute value of x in the function to see if it equals to zero , if not it won't be a factor of the function

\(5 {x}^{3} - 20 {x}^{2} - 5x + 20 = 0 \\ 5(4) ^{3} - 20( {4})^{2} - 5(4) + 20 = 0 \\ 5(64) - 20(16) - 20 + 20 = 0 \\ 320 - 320 - 20 + 20 = 0 \\ 0 = 0\)

so (x-4) is a factor for the function

15:12 is the equivalent ratio of 5:4

Answer:

1 meter is about 3 feet

Step-by-step explanation: To find the approximate length, multiply the length value (1) by 3.281

We can calculate the conversion factors and find that 1 meter is equal to 3.28 feet.

When translating measures of a given amount between different units, multiplicands are often used.

These variables change the value of the measured quantity without altering its consequences.

A quantity that is multiplied or split by two different sets of units is known to as a conversion factor.

If the conversion is required, it must be carried out using the appropriate conversion factor to produce an equivalent value.

For instance, when translating from inches and feet, 12 inches equals one foot.

So, we are given:
1 meter

And while studying conversion factors, we know that:

1 meter = 3.28 feet

Therefore, we can calculate the conversion factors and find that 1 meter is equal to 3.28 feet.

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Answer:

D, 3/13 simplified from 12/52

Step-by-step explanation:

the reason that it would be 3/13 is because its simplified from 12/52, which is 12 of the possibly of face cards over 52 the amount of cards in the deck, so the possibility was 12/52, but simplified would be 3/13 ! love you !

                     BE SAFE !!!!                      HOPE THIS HELPS!!!!

                                              <3 <3 <3 <3 <3

The probability that the card drawn is a face card is 3/13.

Option D is the correct answer.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

We have,

There are 3 face cards.

Number of jack = 4

Number of queen = 4

Number of king = 4

Total number of cards = 52

The probability that the card drawn is Jack.

= 4/52

The probability that the card drawn is Queen.

= 4/52

The probability that the card drawn is King.

= 4/52

The probability that the card drawn is a face card.
= 4/52 + 4/52 + 4/52

= 3 x 4/52

=  3 x 1/13

= 3/13

Thus,

The probability that the card drawn is a face card is 3/13.

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Answer:

B and D

Step-by-step explanation:

Given

x² - x - 12

Consider the factors of the constant term (- 12) which sum to give the coefficient of the x- term (- 1)

The factors are - 4 and + 3 , since

- 4 × 3 = - 12 and - 4 + 3 = - 1 , then

x² - x - 12 = (x - 4)(x + 3) ← in factored form

Answer:

B and D

Step-by-step explanation:

If you call s=-1 the coefficient of x and p=-12 the constant term, you must find two numbers whose sum is s and product is p.

In this case the two numbers are -4 and +3 in fact:

(-4)+(+3)=-1=s

(-4)*(+3)=-12=p

The polynomial can therefore be written as

(x-4)(x+3)

If you want to check this result, just multiply and get:

(x-4)(x+3)=x^2+3x-4x-12=x^2-x-12