Pre-calculus: Lesson 49 properties of logs - Assess it

Answer:

47 - 6√10

Step-by-step explanation:

Area of a square of side a = a²

Here a = √2 -3√5

Area A = a² = (√2 -3√5)²

We know that (a - b)² = a² -2ab +b²

So

A = (√2 -3√5)²

= (√2)² - 2(√2)(3√5) + (3√5)²

(√2)² = 2   because (√a)² = a

(3√5)² = 3² x (√5)² = 9 x 5 = 45

2(√2)(3√5) = 2 (√2 · 3 · √5)

= 2 x 3 √2 √5

√2√5 = √10

2 x 3 √2 √5 = 2 · 3 · √10 = 6√10

Therefore area = 2 - 6√10 + 45

= 47 - 6√10

Proved that if the function f(x,y)=e^(ax+by) with a^2+b^2=1, then f''(xx)+f''(yy)=1

To find the second partial derivatives of f(x, y), we differentiate f(x, y) twice with respect to x and y:

f'(x, y) = ae^(ax + by) and f''(xx) = a^2e^(ax + by)

f'(x, y) = be^(ax + by) and f''(yy) = b^2e^(ax + by)

Adding these second partial derivatives, we get:

f''(xx) + f''(yy) = a^2e^(ax + by) + b^2e^(ax + by)

Since a^2 + b^2 = 1, we have:

f''(xx) + f''(yy) = e^(ax+by)

But f(x, y) = e^(ax+by), so:

f''(xx) + f''(yy) = f(x, y)

Therefore, we have proved that f''(xx) + f''(yy) = 1.

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Answer:

Using the quadratic formula, we find that:

x = 0.555 or x = 89.445

We can ignore the first solution since it doesn't make sense for the speed of a train. Therefore, the speed of the slower train is approximately 89.445 mph, and the speed of the faster train is approximately 94.445 mph.

So, the answer is:

The speed of the slower train is approximately 89.445 mph, and the speed of the faster train is approximately 94.445 mph.NOT SURE

Step-by-step explanation:

Answer: One third

Step-by-step explanation: If you blow up a third of the balloons then you've blown 4 up because 12 divided by 3 is 4. Then you add the 5 balloons your friend blew up leaving you with 9 blown-up balloons in total. 9 blown-up balloons out of 12 is 2 thirds of the balloons leaving you with only one-third left to blow up.

Answer:

1/4

I hope this helps you <3

The probabilities of the events are:

a) P(A and B) = 0

b) P(A or B)  = 0.9

c) P(notA) = 0.55

d) P(notB) = 0.55

e) P(not(A or B)) = 0.1

f) P(A and (not B)) = 0.45

Probability is the measure of the likelihood or chance of an event occurring. It deals with the study of random events and the analysis of their outcomes.

Since A and B are mutually exclusive, they cannot happen at the same time. This means that P(A and B) = 0.

a) P(A and B) = 0

b) P(A or B) = P(A) + P(B)

P(A or B) = 0.45 + 0.45 = 0.9

c) P(notA) = 1 - P(A)

P(notA) = 1 - 0.45 = 0.55

d) P(notB) = 1 - P(B)

P(notB) = 1 - 0.45 = 0.55

e) P(not(A or B)) = 1 - P(A or B)

P(not(A or B)) = 1 - 0.9 = 0.1

f) P(A and (not B)) = P(A) - P(A and B)

P(A and (not B)) = 0.45 - 0 = 0.45

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The interpretation for this interval is that we can be 98% confident that the population mean amount of syrup dispensed is between 1.092 and 1.104 fluid ounces.

A. The data collected in this situation is quantitative.
B. The parameter of interest in this situation is the population mean amount of syrup dispensed.
C. We know that the sample mean is normally distributed because the population of syrup content (fluid ounces) is assumed to be normally distributed with a standard deviation of 0.016 fluid ounces.
D. The appropriate critical value (z α/2 ​ ) for a 98% confidence interval for population mean in this situation is 2.05.
E. Using the appropriate formula, the 98% (two-sided) confidence interval for the population mean amount of syrup dispensed is (1.092, 1.104).
F. The interpretation for this interval is that we can be 98% confident that the population mean amount of syrup dispensed is between 1.092 and 1.104 fluid ounces.

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The height of the cone is 11 mm and the slant height of the cone is 23 mm and the ant crawls 31 mm to get from the base of the cone to the top of the hill.

What is the volume of the right circular cone?

A right circular cone is a three-dimensional geometric shape that has a circular base and a vertex that is located directly above the center of the base. The axis of the cone is a line that passes through the vertex and the center of the base.

The volume of a right circular cone can be found using the formula:

[tex]V = (1/3)\pi r^2h[/tex]

where V is the volume of the cone, r is the radius of the circular base, and h is the height of the cone.

1) To find the height of the cone, we need to use the formula for the volume of a cone, which is [tex]V = (1/3)\pi r^2h[/tex]. Rearranging this formula, we get:

[tex]h = 3V / \pi r^2[/tex]

Substituting the given values, we get:

[tex]h = 3(8792) / (3.14)(20^2)[/tex]

h ≈ 11

Therefore, the height of the cone is approximately 11 mm.

2) To find the slant height of the cone, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In the case of the cone, the slant height is the hypotenuse of a right triangle formed by the height and the radius of the base. Therefore, we have:

[tex]s^2 = r^2 + h^2[/tex]

Substituting the values we found in part 1, we get:

[tex]s^2 = 20^2 + 11^2[/tex]

s ≈ 23

Therefore, the slant height of the cone is approximately 23 mm.

3) To find the distance the ant crawls to get from the base of the cone to the top of the hill, we can use the Pythagorean theorem again. This time, we need to find the distance along the slant height from the base of the cone to the top of the hill. This distance is the hypotenuse of a right triangle formed by the distance from the base to the apex (which is the height of the cone) and the radius of the base. Therefore, we have:

[tex]d^2 = r^2 + s^2[/tex]

Substituting the values we found in parts 1 and 2, we get:

[tex]d^2 = 20^2 + 23^2[/tex]

d ≈ 31

Therefore, the ant crawls approximately 31 mm to get from the base of the cone to the top of the hill.

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Therefore , the solution of the given problem of triangle comes out to be angle x has a magnitude of 101°.

A triangle is a polygon since it possesses two or so additional components beyond that. It is plainly rectangular in form. A and B are the only two of either a triangle's three edges that can distinguish it from a typical triangle. When bounds are still not precisely collinear, Euclidean geometry yields a single region rather than a cube. Three edges and three angles are what make a triangle.

Here,

Because a triangle's angles sum up to 180 degrees, we can determine the size of angle x by deducting the other two angles from 180 degrees:

=> x = 180° - 35° - 44°

=> x = 101°

As a result, angle x has a magnitude of 101°.

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The  sample size of the label needed to fit around the tin of soup is 24.5 cm, calculated by multiplying the diameter (7.5 cm) by pi (3.14) and adding 1 cm for overlap.

In order to calculate the size of the label to fit around the tin of soup, you will need to measure the circumference of the can. The circumference of the can is the total length of the can along the curved edge. To do this, you will need to measure the diameter of the can, which in this case is 7.5 cm, and then multiply it by pi (3.14). This will give you the circumference of the can, which is 23.5 cm. To find the size of the label, you will need to add 1 cm onto the circumference of the can to allow for a small overlap. This will give you a label size of 24.5 cm.

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Answer:

3.6% per annum

Step-by-step explanation:

The formula for accrued amount on a loan(value of loan) is given by the formula if compounded yearly is

[tex]A = P\left(1 + r)^t[/tex]

where

P = original loan amount

r = interest rate in decimal

t = number of years

We are given that the original loan amount
P = £5500

After 1 year the accrued amount
[tex]A = 5500(1 + r)^1[/tex]

and we know that after 1 year, the value of the loan is £5698

Therefore

5698 = 5500(1 + r)

1+ r = 5698/5500 = 1.036

r= 1.036 - 1 = 0.036

In percentage terms 0.036 = 0.036 x 100 = 3.6%

The zeroes of the polynomial f(x) = x⁴ - 2x³ - 6x² + 22x - 15 are x = 1, x = 3, x = -3

Polynomials can be classified by degree, which is the highest power of the variable in the polynomial. For example, the polynomial f(x) = 3x^4 - 2x^3 + x^2 - 5x + 2 is a fourth-degree polynomial, as the highest power of x is 4.

Polynomials can be added, subtracted, multiplied, and divided using a variety of techniques, including factoring, synthetic division, and long division. They can also be evaluated at specific values of the variable, which can be useful in solving problems or analyzing data.

We can find the zeroes of the polynomial function f(x) = x⁴ - 2x³ - 6x² + 22x - 15 by factoring it or by using numerical methods such as the Newton-Raphson method or the bisection method. Here, we will use the factoring method.

First, we can observe that x = 1 is a zero of the polynomial, since f(1) = 1 - 2 - 6 + 22 - 15 = 0. Therefore, we can factor out (x - 1) from the polynomial:

f(x) = x⁴ - 2x³ - 6x² + 22x - 15

= (x - 1)(x³ - x² - 7x + 15)

Therefore, we have:

x³ - x² - 7x + 15 = (x - 1)(x² - 9)

= (x - 1)(x - 3)(x + 3)

Therefore, the zeroes of the polynomial f(x) = x⁴ - 2x³ - 6x² + 22x - 15 are:

x = 1, x = 3, x = -3

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Answer:

Slope = (-3/2)

Step-by-step explanation:

Slope(m) = (Y2 - Y1) / (X2 - X1)

= (0-3) / (2-0)

= (-3/2) Is the ans....

Infinite solution

The answer is 0. Therefore, there is Infinite solutions

[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ V=3.6\\ h=4.2 \end{cases}\implies 3.6=\cfrac{\pi r^2(4.2)}{3}\implies 10.8=4.2\pi r^2 \\\\\\ \cfrac{10.8}{4.2\pi }=r^2\implies \sqrt{\cfrac{10.8}{4.2\pi }}=r\hspace{5em}\stackrel{\textit{diameter = 2r}}{2\sqrt{\cfrac{10.8}{4.2\pi }}} ~~ \approx ~~ \text{\LARGE 1.8}[/tex]

The two z-scores that separate the middle 87.4% of the distribution from the area in the tails of the standard normal distribution are -1.46 and 1.46.

To find these z-scores, we need to first determine the area in the tails of the distribution. Since the middle 87.4% of the distribution is separated from the tails, the area in the tails is 100% - 87.4% = 12.6%. This means that there is 12.6% / 2 = 6.3% in each tail.

Next, we can use a standard normal table or a calculator to find the z-scores that correspond to the 6.3% in the tails. We want to find the z-scores that have an area of 0.063 to the left and to the right of them.

According to a standard normal table, the z-score that has an area of 0.063 to the left of it is -1.46, and the z-score that has an area of 0.063 to the right of it is 1.46.

Therefore, the two z-scores that separate the middle 87.4% of the distribution from the area in the tails of the standard normal distribution are -1.46 and 1.46.

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The dilation is an enlargement, and Tashi is correct.

To determine whether the dilation (scaling) is a reduction or an enlargement, we need to compare the distances between corresponding points on the original figure and the dilated figure.

Suppose a point P(x,y) is on the original figure. After the dilation with the rule (6/5x, 6/5y), the corresponding point P'(x', y') will be:

P' = (6/5x, 6/5y)

The distance between point P and the origin is given by:

d(P) = √(x² + y²)

The distance between point P' and the origin is:

d(P') = √((6/5x)² + (6/5y)²)

= 6/5 * √(x² + y²)

Since the dilation multiplies each coordinate by a factor of 6/5, the distance between any point and the origin is multiplied by a factor of 6/5.

Therefore, if 6/5 < 1, then the distances between corresponding points are reduced, and the dilation is a reduction. If 6/5 > 1, then the distances are increased, and the dilation is an enlargement.

In this case, we have:

6/5 > 1

Therefore, the dilation is an enlargement, and Tashi is correct.

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In the year 2000, there were 44 hazardous waste sites in State Y.

Hazardous materials and activities are those that pose a risk to human health, safety and the environment. Examples of hazardous materials include chemicals, flammable and combustible substances, radioactive materials, explosives, biological agents and other substances that may cause harm. Examples of hazardous activities include working with hazardous materials, operating machinery, working with electricity, and performing construction activities. It is important to understand the potential risks associated with hazardous materials and activities in order to protect oneself and others from potential harm.

n + 8 = 2(34 - 8)
n + 8 = 2(26)
n + 8 = 52
n = 44
In the year 2000, there were 44 hazardous waste sites in State Y.

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The values in parts (a) and (b) are numerical summaries of the population.

Regarding the question on baseball players' salaries in 2015, the mean salary for all 868 professional baseball players is 3.856 million dollars and the standard deviation is 1.589 million dollars. The values in parts (a) and (b) are numerical summaries of the population.

The shape of the distribution of salaries for professional baseball players in 2015 is positively skewed, with the majority of the salaries clustering at the lower end of the salary range and a few salaries on the upper end. There is only one mode to the distribution.

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Answer:

a.4

Step-by-step explanation: You first work on the parentheses( 7+9)=16. Then you multiply m and 16. Which is 0.5(16) or 16/2=8 now it’s 8/2 which is 4 .

Answer:

B

Step-by-step explanation:

In answer choice B, there is the same number of medium bottles as the number of medium bottles that you are given in the question.

Because you know that there are 93 fluid ounces in six small bottles and four medium bottles, you know that any different amount of fluid ounces with the answer choice would be because of the increase in small bottles.

By having a consistent change with only one variable (which is the small bottle) you can find the rate of change. Once you figure out how many fluid ounces are in the small bottles, you can find the total amount of fluid ounces that are in the small bottles. The amount that remains is the total fluid ounces for medium bottles.

From the given data, the probability of Z less than 1.56, greater than 1.56, between 1.56 and 1.81 and less than 1.56 or greater than 1.81 is 0.9406, 0.0359, 0.0235 and 0.9765 respectively.

a. From the cumulative standardized normal distribution table, we can look up the value 1.56 in the left column and find the corresponding value of 0.9406 in the intersecting row. Therefore, the probability that Z is less than 1.56 is 0.9406.

b. From the cumulative standardized normal distribution table, we can look up the value 1.81 in the left column and find the corresponding value of 0.9641 in the intersecting row. Since we want the probability that Z is greater than 1.81, we subtract this value from 1 to get 1 - 0.9641 = 0.0359 (rounded to four decimal places as needed). Therefore, the probability that Z is greater than 1.81 is 0.0359.

c. To find the probability that Z is between 1.56 and 1.81, we need to subtract the probability that Z is less than 1.56 from the probability that Z is less than 1.81. From the table, we know that the probability that Z is less than 1.56 is 0.9406 and the probability that Z is less than 1.81 is 0.9641. Therefore, the probability that Z is between 1.56 and 1.81 is 0.9641 - 0.9406 = 0.0235 (rounded to four decimal places as needed).

d. The probability that Z is less than 1.56 or greater than 1.81 is the sum of the probabilities that Z is less than 1.56 and Z is greater than 1.81, since these events are mutually exclusive. From parts (a) and (b), we know that the probability that Z is less than 1.56 is 0.9406 and the probability that Z is greater than 1.81 is 0.0359. Therefore, the probability that Z is less than 1.56 or greater than 1.81 is 0.9406 + 0.0359 = 0.9765 (rounded to four decimal places as needed).

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The Pacific Ocean is roughly 100 times bigger than the Gulf of Mexico.

Ocean is the largest and most prominent feature of the Earth's surface. It covers more than 70% of the planet and contains 97% of its water. Oceans are vital to life on Earth, providing food and resources, regulating the climate, and supporting a vast array of marine life. They are home to a variety of habitats including coral reefs, deep sea trenches, and shallow coastal estuaries.

The Pacific Ocean covers an area of roughly 63.8 million square miles (165.2 million square kilometers), while the Gulf of Mexico covers an area of only 615,000 square miles (1.6 million square kilometers). This means that the Pacific Ocean is roughly 6 x 105 (or 600,000) times bigger than the Gulf of Mexico.

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Without a clear pattern or rule from the given examples, it is difficult to determine the value of 1*7 in this problem.

This problem appears to involve some sort of pattern or rule that relates the multiplication of two numbers to a resulting value. Let's examine the given examples to try to identify the pattern:

5*3=4: The product of 5 and 3 is 15, but the answer is 4. This means that some sort of operation was performed on 15 to produce 4. One possibility is that the digits of 15 were added together: 1 + 5 = 6, and then the result was subtracted from 10: 10 - 6 = 4. So, this pattern involves adding the digits of the product and subtracting the result from 10.

2*8=2: The product of 2 and 8 is 16, but the answer is 2. Again, we need to perform an operation on 16 to get 2. One possibility is to divide 16 by the sum of the digits: 1 + 6 = 7, so 16 / 7 = 2.28 (rounded to 2 decimal places). However, this pattern doesn't quite fit the given examples perfectly, so there may be other rules or variations involved.

6*3=3: The product of 6 and 3 is 18, but the answer is 3. Applying the same approach as before, we could try dividing 18 by the product of the digits: 6 * 3 = 18, so 18 / 18 = 1. However, this pattern doesn't match the previous examples either.

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Answer:

Step-by-step explanation:

Square (top right), rectangle (middle bottom), and trapezoid (bottom right)

if u do it right then u should get this answer.

The correct option for the probability of landing in the color red is:

Experimental: 40%; theoretical: 37.5%

What is  probability?

A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%.

We will get

p = 8/20 = 2/5 = 0.4

In percentage form this is:

p = 100%*0.4 = 40%

While the theoretical probability depends on the quotient between the number of red sections and the total number of sections.

There are 3 red sections and 8 sections in total, so here the probability is:

p = 100%*3/8 = 37.5%

So the correct option is C.

Experimental: 40%; theoretical: 37.5%

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If MNO is NMO, it means that the second and third vertices of the triangle have switched places, but the first vertex remains the same. Therefore, triangle MON is congruent to triangle NMO.

as they have the same side lengths and angles. Specifically, the sides MO, ON, and NM are congruent, and the angles at vertices M, O, and N are also congruent. We can represent this relationship between the two triangles using the symbol ≅, so we can say that triangle MON ≅ triangle NMO. The concept of congruent triangles is a fundamental principle in geometry. Two triangles are said to be congruent if their corresponding sides and angles are equal. When two triangles are congruent, we can say that they have the same size and shape, but they may be oriented differently in space. here are several ways to prove that two triangles are congruent, such as the side-angle-side (SAS), angle-side-angle (ASA), side-side-side (SSS), angle-angle-side (AAS), and hypotenuse-leg (HL) congruence criteria.

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Answer: Therefore, Hamid needs to pay Rs. 1050 as both CGST and SGST.

Step-by-step explanation:

Assuming the GST rate is the same for both the purchase and the sale, we can calculate the GST rate as follows:

GST rate = GST paid / Cost of stationary purchased

= 1800 / Cost of stationary purchased

Since we don't have the cost of stationary purchased, we can't calculate the GST rate or the CGST and SGST amounts directly. However, we know that the GST collected from Nishaben was Rs. 2100, which is equal to the GST paid plus the profit earned on the sale. We can express this mathematically as follows:

GST collected = GST paid + Profit

Profit = GST collected - GST paid

= 2100 - 1800

= Rs. 300

Since the profit earned is Rs. 300, we can find the cost of the stationary purchased by subtracting the profit from the amount collected from Nishaben:

Cost of stationary purchased = GST collected - Profit

= 2100 - 300

= Rs. 1800

Now that we know the cost of the stationary purchased, we can calculate the GST rate as follows:

GST rate = GST paid / Cost of stationary purchased

= 1800 / 1800

= 1

Since the GST rate is 1 (or 100%), we can divide the total GST collected equally between CGST and SGST:

CGST = SGST = GST collected / 2

= 2100 / 2

= Rs. 1050

Therefore, Hamid needs to pay Rs. 1050 as both CGST and SGST.

Answer:

See explanation below

Step-by-step explanation:

1/(sec - tan) - 1/cos = 1/cos - 1(sec + tan)

=> 1/(sec - tan) + 1(sec + tan) = 1/cos + 1/cos

Left side

(sec + tan)/[(sec - tan)(sec + tan)] + (sec - tan)/[(sec - tan)(sec + tan)]

= [sec + tan + sec - tan]/[(sec - tan)(sec + tan)]

= [2sec]/[(sec - tan)(sec + tan)]

since (a+b)(a-b) = a^2 - b^2

=> 2sec/(sec^2 - tan^2)

let's work on the denominator (sec^2 - tan^2)

we know sec = 1/cos and tan = sin/cos

=> (1/cos)^2 - (sin/cos)^2

=> 1/cos^2 - sin^2/cos^2

=> (1 - sin^2)/cos^2 = cos^2/cos^2 = 1

so 2sec/(sec^2 - tan^2) => 2sec/1

=> 2sec

since sec = 1/cos => 2sec = 2(1/cos) = 2/cos

Right side

1/cos + 1/cos = 2/cos