6) If sec theta+tan theta = P. PT sin theta=P^2-1/P^2+1 ...?

We have been given the function [tex]x^{15}-64[/tex]

Convert this expression in difference of cubes form.

[tex]x^{15}-64\\=(x^5)^3-4^3[/tex]

Now, in order to find the factors, we can apply the difference of cubes form, which is given by

[tex]x^3-y^3=(x-y)(x^2+xy-y^2)[/tex]

On applying this formula, we get

[tex](x^5-4)(x^{10}+4x^5+16)[/tex]

Hence, the factors of the given expression are

[tex](x^5-4) \text{ and }(x^{10}+4x^5+16)[/tex]

Out of the given option, A is the correct option.

x^5 − 4 is the required factor.

The factors of the polynomial [tex]x^{15}-64[/tex] are  [tex]x^{5}-4[/tex] and  [tex]x^{10}+4x^{5}+16[/tex].

Given data:

The polynomial equation is represented as [tex]f(x)=x^{15}-64[/tex].

On simplifying the equation:

[tex]f(x)=[x^{5}]^{3}-4^{3}[/tex]

From the difference of cubes:

[tex]x^{3}-y^{3}=(x-y)(x^{2}+xy-y^{2})[/tex]

Substituting the values in the equation:

[tex][x^{5}]^{3}-4^{3}=(x^{5}-4)(x^{10}+4x^{5}+16)[/tex]

Hence, the factors of the equation is [tex]x^{5}-4[/tex].

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The product of the given multiplication is 10000.

Multiplication is when you take one number and add it together a number of times.

Example: 5 multiplied by 4 = 5 + 5 + 5 + 5 = 20.

Given that, multiply 100 x 100,

100 x 100 = 10000

Hence, the product of the given multiplication is 10000.

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Jennifer has 27 nickels and 56 dimes.

Let's use the variables n and d to represent the number of nickels and dimes Jennifer has, respectively.

We can set up two equations based on the given information:

Now, we can solve this system of equations.

We can rewrite equation 1 as n = 83 - d and substitute it into equation 2:

0.05(83 - d) + 0.10d = 6.95

4.15 - 0.05d + 0.10d = 6.95

0.05d = 6.95 - 4.15

0.05d = 2.80

d = 2.80 / 0.05

d = 56

Substituting the value of d back into equation 1, we find n = 83 - 56 = 27.

Therefore, Jennifer has 27 nickels and 56 dimes.

In part (a), the expression for cost as a single fraction is (6x^2 + 900000x) / x. In part (b), we substitute x = 240 and simplify the expression to find the cost for ordering and storing 240 units of the product.

(a) To write the expression for cost as a single fraction, we need to combine the terms in the numerator. The numerator is 6x + 900000 and the denominator is x. To combine the terms, we multiply 6x by x to get 6x^2, and then add 6x^2 + 900000x in the numerator. Therefore, the expression for cost is (6x^2 + 900000x) / x.

(b) To determine the cost for ordering and storing x = 240 units of this product, we substitute x = 240 into the expression for cost. Plugging in x = 240, we have (6(240)^2 + 900000(240)) / 240. Simplifying this expression gives us the cost for ordering and storing 240 units of the product.

The cost function C = 6x + 900000/x can be written as a single fraction as C = (6x² + 900000) / x. When x = 240, the cost for ordering and storing the units is calculated to be $5190.

The cost C of ordering and storing x units of a product is given by the formula C = 6x + 900000/x. To write this expression as a single fraction, we can find a common denominator and combine the two terms:

C = (6x²+ 900000) / x

For part (b), to determine the cost for ordering and storing x = 240 units of this product, we substitute 240 for x in the expression:
C = (6(240)² + 900000) / 240

Which simplifies to:
C = (6(57600) + 900000) / 240

C = (345600 + 900000) / 240

C = 1245600 / 240

C = 5190

Therefore, the cost for ordering and storing 240 units of this product is $5190.

Answer:

B. 29%

Step-by-step explanation:

if there is a 30% chance of sun tomorrow and a 20% chance of wind and no sun.

what is the probability that it is windy, given that it is not sunny.

A: it is not sunny

P(A)=0.7    (since, chance of being sunny=30% then chance of not being sunny=70%)

B:it is windy

A∩B:it is windy and not sunny

P(A∩B)=0.2  (chance=20%)

B/A:it is windy given that it is not sunny

Baye's theorem states that:

P(A∩B)=P(B/A)×P(A)

0.2=P(B/A)×0.7

⇒ P(B/A)= 2/7=0.286

             ≈ 29%

Hence, probability that it is windy, given that it is not sunny is:

 B.  29%

Answer:

x=4

Step-by-step explanation:

The value of x that  makes the triangle equilateral is 4

For an equilateral triangle, all three sides must be equal in length. In this case, the sides are represented by 3x, 2x + 4, and 5x - 8.

To find the value of x that makes the triangle equilateral, set up an equation:

3x = 2x + 4 = 5x - 8

Now, solve for x:

3x = 2x + 4.

Subtract 2x from both sides:

x = 4

Now, check if x = 4 satisfies the other two parts: 2x + 4 and 5x - 8.

For 2x + 4:

2(4) + 4 = 8 + 4 = 12

For 5x - 8:

5(4) - 8 = 20 - 8 = 12

Since all three expressions are equal to 12, the value of x = 4 makes the triangle equilateral.

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The rectangle with the maximum area that can be inscribed within the parabola y=7-x^2 has dimensions length 2√(7/3) and height 14/3.

The problem is asking to find the dimensions of a rectangle inscribed within a parabola with maximum area. This is an example of an optimization problem in calculus. Here, the parabola given by y=7-x^2 suggests that the rectangle's corners on the x-axis lie at (-x, 0) and (x, 0). The corners on the parabola would be located at (-x, 7-x^2) and (x, 7-x^2). The dimensions of the rectangle would be 2x for the length (base) and 7-x^2 for the height.

For the maximum area, we need to take the derivative of the area function A = length*width = (2x)*(7-x^2) and set it equal to zero, thus defining the critical points.

The derivative of A is given by A' = 14 - 6x^2. Setting A' equal to zero gives: x = √(14/6) which simplifies to x = √(7/3).

Finally, as we are looking for the dimensions of the rectangle, the length (base) is 2*x = 2√(7/3) and the height is given by y = 7 - x^2 = 7 - (7/3) = 14/3.

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Answer:

Kelli's height = [tex]\frac{3}{4}x[/tex]

Step-by-step explanation:

Here, x represents the Ted's height .

As per the statement:

Kelli is [tex]\frac{3}{4}[/tex] as tall as her brother Ted.

We have to find an expression to describe Kelli's height.

then;

[tex]\frac{3}{4}[/tex] as tall as her brother Ted means [tex]\frac{3}{4}x[/tex]

⇒Kelli's height =  [tex]\frac{3}{4}x[/tex]

Therefore, an expression to describe Kelli's height is, [tex]\frac{3}{4}x[/tex]

To convert 2x = 2y into polar form, replace x with r cos(theta) and y with r sin(theta), then simplify, resulting in the polar equation cos(theta) = sin(theta) or theta = pi/4 + kpi, where k is an integer.

To convert the equation 2x = 2y to polar form, we need to use the relationship between Cartesian coordinates (x, y) and polar coordinates (r, (theta)). The transformations are given by x = r cos(theta) and y = r sin(theta).

Starting with our original equation 2x = 2y, we can substitute the transformations into the equation, which gives us 2(r cos(theta)) = 2(r sin(theta)). We can divide both sides by 2 to simplify, which results in r cos(theta) = r sin(theta). Since r cannot be 0 (because then both x and y would be 0, which does not satisfy the original equation), we can divide both sides by r, leading to cos(theta) = sin(theta). This equation implies that tan(theta) = 1. When tan(theta) equals 1, theta can be written as theta = arctan(1) or theta = pi/4 + kpi, where k is an integer representing the multiple full rotations around the circle.

The probability that a dessert chosen at random contains nuts given that it contains chocolate is approximately 0.34.

To find the probability that a dessert chosen at random contains nuts given that it contains chocolate, we can use the formula for conditional probability:

P(N|C) = P(C and N) / P(C)

We are given that the probability that a dessert contains chocolate is 73% or 0.73, and the probability that a dessert containing chocolate also contains nuts is 25% or 0.25. Therefore, P(N|C) = 0.25 / 0.73 ≈ 0.34, rounded to the nearest tenth of a percent.

The probability that a dessert contains nuts given it contains chocolate is approximately 34.3%.

Probability of a dessert containing chocolate = 73% (0.73)
Probability of a dessert containing nuts given it contains chocolate = 25% (0.25)
Probability that a dessert contains nuts given it contains chocolate
Let C be the event that the dessert contains chocolate and N be the event that it contains nuts.

Therefore, the probability that a dessert chosen at random contains nuts given that it contains chocolate is approximately 34.3%.

Answer:

None of the options is correct

Step-by-step explanation:

It is important to know that he expression a < x < b means that x is greater than a and less than b. So, for finding the correct value let's analyse each option.

A. 4 < 50 < 5

50 is greater than 4 but it is not less than 5, so, option A is incorrect.

B. 7 < 50 < 8

50 is greater than 7 but it is not less than 8, so, option B is incorrect.

C. 8 < 50 < 9

50 is greater than 8 but it is not less than 9, so, option C is incorrect.

D. 10 < 50 < 11

50 is greater than 10 but it is not less than 11, so, option D is incorrect.

Thus, none of the options is correct.

The correct inequality that should show the value of 50 is 49 < 50 < 51, but based on the provided options and context, all are incorrect. If choosing the closest correct answer, it would be 'C. 8 < 50 < 9', acknowledging that it is still incorrect because 50 is not less than 9. Hence, option C.

The student's question is about determining the correct inequality that shows the value of 50 within two consecutive integers. When comparing numbers, we need to find two consecutive integers between which the number 50 falls. The correct inequality that illustrates this is D. 10 < 50 < 11, as 50 is greater than 10 and less than 11. Considering context, this seems to be a typo, since 50 cannot be between 10 and 11. The correct answer should logically be none of the above, as 50 is actually between 49 and 51, making all the provided options incorrect. However, if the intention was to find which of the provided options is closest to being correct, then the answer would be C. 8 < 50 < 9, as 50 is greater than 8 but certainly not less than 9. It is important to note that this question involves an error in the provided options.

The diagonals of a square are perpendicular to each other. Thus, the correct option is A.

It is a polygon with four sides. The total interior angle is 360 degrees. A square's opposite sides are parallel and equal, and each angle is 90 degrees. Its diagonals are all the same length and intersect in the center.

If the angle between the diagonals of the quadrilateral is not equal to the right angle, then the quadrilateral is a rectangle, parallelogram, or trapezoid.

If the angle between the diagonals of the quadrilateral is equal to the right angle, then the quadrilateral is a rhombus and square.

The diagonal of the square intersects at the right angle. Then the diagonals of a square are perpendicular to each other.

Thus, the correct option is A.

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Answer:

3

Step-by-step explanation:

The relative growth rate per year is approximately 22521.43 people per year. The U.S. population at 3:45 PM Boston time on May 11, 2001, will be approximately 275,333,002 people.

The relative growth rate per year can be found by converting the rate of 1 person every 14 seconds to the rate per year. There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day, so there are 60 * 60 * 24 = 86400 seconds in a day. There are 365 days in a year, so the rate per year is (1 person / 14 seconds) * (86400 seconds / 1 day) * (365 days / 1 year) = 22521.4286 people per year. Therefore, the relative growth rate per year is approximately 22521.43 people per year.

To determine the U.S. population at 3:45 PM Boston time on May 11, 2001, we need to calculate the number of seconds between May 11, 1993, and May 11, 2001, at 3:45 PM. There are 365 days in a year and 24 hours in a day, so there are 365 * 24 = 8760 hours in a year. Since May 11, 1993, to May 11, 2001, is 8 years, there are 8 * 8760 = 70080 hours between the two dates. There are 60 minutes in an hour and 60 seconds in a minute, so there are 60 * 60 = 3600 seconds in an hour. Therefore, there are 70080 * 3600 = 252288000 seconds between the two dates. The population increase rate is 1 person every 14 seconds, so the population increase during this time period is 252288000 / 14 = 18020571.43 people. Adding this to the initial population of 257,313,431 gives a total population of 257,313,431 + 18020571.43 = 275,333,002.43. Therefore, the U.S. population at 3:45 PM Boston time on May 11, 2001, will be approximately 275,333,002 people.

The first option is correct:

1.) if to rays are opposite.

Let

n--------> number of monthly payments

x-------> amount of monthly payments with Kristal's credit card

y-------> charges for every payment made with a credit card

T-------> total cost of the refrigerator

we know that

[tex] n=16\\x=\$112.75\\ y=\$1.25\\[/tex]

the total cost of the refrigerator is equal to

[tex] T=n(x+y) [/tex]

substitute the values in the formula

[tex] T=16*(112.75+1.25) [/tex]

[tex] T=16*114 [/tex]

[tex] T=\$1,824 [/tex]

therefore

the answer is

the total cost of the refrigerator is equal to [tex] \$1,824 [/tex]

Method 1

Applying the Pythagorean Theorem

we know that

[tex]10^{2}= 5^{2} +a^{2}[/tex]

Solve for a

[tex]100= 25 +a^{2}[/tex]

[tex]a^{2}=100-25[/tex]

[tex]a^{2}=75[/tex]

[tex]a=\sqrt{75}=5 \sqrt{3}\ units[/tex]

therefore

the answer is

the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]

Method 2

we know that

[tex]sin(60\°)=\frac{\sqrt{3}}{2}[/tex] -------> equation A

and

in this problem

[tex]sin(60\°)=\frac{a}{10}[/tex] --------> equation B

equate equation A and equation B

[tex]\frac{\sqrt{3}}{2}=\frac{a}{10}\\\\a=\frac{10\sqrt{3}}{2}\\\\a=5 \sqrt{3}\ units[/tex]

therefore

the answer is

the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]

Answer:

Triangle Proportionality Theorem

Step-by-step explanation:

Just took the test xD

Abdul can use the Pythagorean Theorem to calculate the straight-line distances on his map, utilizing the formula a² + b² = c² for right triangles created by the intersections of his neighborhood's streets.

To determine the remaining distances for his map, Abdul can apply the Pythagorean Theorem. The streets in question form right angles at their intersections, as implied by the parallel streets mentioned. Thus, if Abdul needs to find the distance between two points that form a right angle with another point (essentially creating a right triangle), he can use the formula a² + b² = c², where a and b are the legs (sides forming the right angle) of the triangle and c is the hypotenuse (the side opposite the right angle).

For example, if Abdul wants to find the straight-line distance (hypotenuse) between his home and the high school, and he knows the distance to the middle school (3 miles) and the distance from the middle school to the high school (6 miles), he can use the Pythagorean Theorem to calculate this. Assuming his home, the middle school, and the high school form a right triangle, the formula becomes: 3² + 6² = c², which after calculations gives c =
√(3² + 6²).

Therefore, the correct answer is pythagoras theorm.