five consecutive multiples of 11 have a sum of 220. what is the greatest of these numbers

A. 33
B. 44
C. 55
D. 66

[tex]\frac{3}{4} = log_{2}(\sqrt[4]{8} )\\-4 = log_{3} \frac{1}{81} \\-6= -3log_{5} 25\\\frac{1}{3} = log_{6} (\sqrt[3]{6} )[/tex]

Here's how you solve it!

[tex]log_{2} \sqrt[4]{8}[/tex]

Write it in exponential form

[tex]log_{2} (2 \frac{3}{4} )[/tex]

Then simplify

[tex]\frac{3}{4}[/tex]

[tex]log_{3} \frac{1}{81}[/tex]

Write in exponential form

[tex]log_{3} (3^{-4} )[/tex]

Simplify

-4

[tex]-3log_{5} 25[/tex]

Write in exponential form

[tex]-3log_{5} (5^{2} )[/tex]

Simplify

-3 * 2 = -6

-6

[tex]log_{6} \sqrt[3]{6}[/tex]

Write in exponential form

[tex]log_{6} (6\frac{1}{3} )[/tex]

Simplify

[tex]\frac{1}{3}[/tex]

Hope this helps! :3

The problem involves simplifying mathematical expressions, through steps as prescribed by BIDMAS/PEDMAS rules. Start by addressing anything within parentheses, follow through with multiplication or division, and finally handle addition or subtraction.

This question involves the process of mathematical simplification of expressions. To solve this, you will first need to perform any calculations within the parentheses, then handle any multiplication or division from left to right, lastly address any addition or subtraction, also from left to right (also known as the order of operations or BIDMAS/PEDMAS). For example, if you have an expression like '2(3+4)': First, process the operation within the parentheses, in this case, it's a sum so you have '2*7', resulting in '14'. This is considered the simplified version of your expression.

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Answer:

  6 2/3 minutes

Step-by-step explanation:

Their rates in "jobs per hour" are ...

  (60 min/h)/(15 min/job) = 4 jobs/h

and

  (60 min/h)/(12 min/job) = 5 jobs/h

So, their combined rate is ...

  (4 jobs/h) + (5 jobs/h) = 9 jobs/h

The time required (in minutes) is ...

  (60 min/h)/(9 jobs/h) = (60/9) min = 6 2/3 min

Working together, it will take them 6 2/3 minutes.

To find out how long it would take the two grandparents to clean the playroom together, we can use the concept of rates and set up an equation. Solving the equation, we find that it would take them 9 minutes to clean the playroom if they work together.

To solve this problem, we can use the concept of rates to find the combined rate at which the two grandparents clean. Let's assign the variable x to represent the time it takes for them to clean together.

The rate at which the first grandparent cleans is 1/15th of the playroom per minute, while the rate at which the second grandparent cleans is 1/12th of the playroom per minute. The combined rate when they work together is the sum of their individual rates, which is given by the equation (1/15)+(1/12)=(1/x).

To solve this equation, we can find a common denominator of 60 to simplify the equation to 4/60+5/60=1/x. Adding the fractions gives us 9/60=1/x. Multiplying both sides of the equation by 60 gives us 9=x. Therefore, it would take the two grandparents 9 minutes to clean the playroom if they work together.

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The graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].

Further explanation:  

The height of the wave is given by the equation as follows:  

[tex]y=asin\left(\dfrac{2\pi t}{T}\right)[/tex]                          ......(1)

Here, [tex]a[/tex] is amplitude, [tex]T[/tex] is period of wave in second and [tex]t[/tex] time in seconds.  

The height [tex]y[/tex] of the wave is given as 2 feet and time [tex]t[/tex] is given as 5 seconds.  

Substitute 2 for [tex]y[/tex] and 5 for [tex]t[/tex] in equation (1).  

[tex]2=asin\left(\dfrac{2\pi \times5}{T}\right)\\2=asin\left(\dfrac{10\pi}{T}\right)\\\dfrac{2}{a}=sin\left(\dfrac{10\pi}{T}\right)[/tex]

The above eqution is valid only for [tex]a\geq 2[/tex] because the maximum value of the term [tex]sin(10\pi /T)[/tex] is 1.  

If [tex]T[/tex] is the time period then in [tex]T[/tex] seconds the graph will hit at least 2 times over 2 feet for [tex]a>2[/tex].

T seconds[tex]\rightarrow[/tex]2 hits

1 seconds [tex]\rightarrow[/tex] [tex]\dfrac{2}{T}[/tex] hits

5 seconds [tex]\rightarrow\dfrac{2\times5}{T}[/tex]

5 seconds [tex]\rightarrow[/tex] [tex]\dfrac{10}{T}[/tex]

If [tex]T[/tex] is time period in 5 seconds then the graph will hit [tex][10/T][/tex] times in interval 0 to [tex]2\pi[/tex].

Thus, the graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].

Learn more:  

1. What is the y-intercept of the quadratic function f(x) = (x – 6)(x – 2)? (0,–6) (0,12) (–8,0) (2,0)  

https://brainly.com/question/1332667  

2. Which is the graph of f(x) = (x – 1)(x + 4)?  

https://brainly.com/question/2334270

Answer details:  

Grade: High school.  

Subjects: Mathematics.  

Chapter: function.  

Keywords: Function, wave equation, height, amplitude, equation, period, periodic function, y=asin(2pit/T), frequency, magnitude, feet, height, time period, seconds, inequality, maximum value, range, harmonic motion, oscillation, springs, strings, sonometer.

Answer:

C(x)= 41x + 680

Step-by-step explanation:

If the fixed cost is 680, that will apply regardless of how many jackets the company makes for you. The number of jackets is unknown.  However, we know that the cost of producing a single jacket is 41, so we can represent that expression as 41x.  Putting those things together gives us a function of the cost:

C(x) = 41x + 680

The equation to determine the total cost encountered by Marty's Tee Shirt & Jacket Company in producing x jackets is C(x) = 680 + 41x.

To determine the total cost, C(x), encountered by Marty's Tee Shirt & Jacket Company in producing x jackets, we need to consider both the fixed costs and the variable costs. The fixed costs, which are $680, are incurred regardless of the number of jackets produced. The variable costs, which are $41 per jacket, increase with each additional jacket produced. So the equation to calculate the total cost is:

C(x) = fixed costs + (variable costs per jacket) * x

Substituting the given values, the equation becomes:

C(x) = 680 + 41x

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Answer:

Step-by-step explanation:

1. -2y+7 < 1

Add 2y-1:

  6 < 2y

Divide by 2:

  3 < y

__

2. 4y +3 < -5

Subtract 3:

  4y < -8

Divide by 4:

  y < -2

_____

These are graphed on the number line with open circles because y=-2 and y=3 are not part of the solution set.

Answer:

Step-by-step explanation:

[tex](1)\\\\-2y+7<1\qquad\text{subtract 7 from both sides}\\-2y+7-7<1-7\\-2y<-6\qquad\text{change the signs}\\2y>6\qquad\text{divide both sides by 2}\\\boxed{y>3}\\\\(2)\\\\4y+3<-5\qquad\text{subtract 3 from both sides}\\4y+3-3<-5-3\\4y<-8\qquad\text{divide both sides by 4}\\\boxed{y<-2}\\\\\text{From (1) and (2) we have:}\ y<-2\ or\ y>3[/tex]

[tex]<,\ >-\text{op}\text{en circle}\\\leq,\ \geq-\text{closed circle}[/tex]

Answer:

  3x ≥ -6

Step-by-step explanation:

"No less than" means "greater than or equal to". An appropriate translation of the problem statement is ...

  3x ≥ -6

Answer:

3x ≥ -6

Step-by-step explanation:

The the inequality that could be used to solve three times a number is no less than negative six is 3x ≥ -6.

Answer:

Question 1: the slope is -6

Question 2: the first choice is the one you want

Step-by-step explanation:

For the first one, I can't tell what fraction is on the left side with the y, but it doesn't matter.  To me it looks like 1/2, but like I said, it won't change or affect our answer regarding the slope.  That number has nothing to do with the slope.  

In order to determine the slope of that line that is currently in point-slope form, we need to change it to slope-intercept form.  Another expression for slope-intercept form is to solve it for y.  Doing that:

[tex]y - \frac{1}{2}=-6x-42[/tex]

Now we can add 1/2 to both sides.  That gives us the slope-intercept form of the line:

[tex]y=-6x- \frac{83}{2}[/tex]

The form is y = mx + b, where the number in the "m" place is the slope.  Our slope is -6.

For the second one, we will sub in the x coordinate in a pair for x in the equation of the line and do the same for y to see if the left side equals the right side.  The answer is [tex](\frac{2}{9},-7)[/tex] and I'll show you why.  I will also show you how another point DOESN'T work in the equation.  Filling in 2/9 for x and -7 for y:

[tex]-7+7=-3( \frac{2}{9} -\frac{2}{9})[/tex] which simplifies to

0 = -3(0) so

0 = 0 and this is true.

The other point I am going to use in exactly the same process is (-3, -7) since it doesn't have fractions in it.  First I'm going to distribute the -3 into the parenthesis to get:

[tex]y+7= -3 x + \frac{6}{9}[/tex]

Subbing in -3 for x and -7 for y:

[tex]-7+7=-3( -3) +\frac{6}{9}[/tex]

As you can see, the left side equals 0 but the right side does not.  If the lft side doesn't equal the right side, then the expression is not true, so the point is not on the line.

Answer:

  18.8 cm²

Step-by-step explanation:

Sometimes, as here, when the problem is not carefully constructed, the answer you get depends on the method you choose for solving the problem.

Following directions

Using the formula ...

  Area = (1/2)ab·sin(C)

we are given the values of "a" (BC=5.9 cm) and "b" (AC=7.2 cm), but we need to know the value of sin(C). The problem statement tells us to round this value to the nearest hundredth.

  sin(C) = sin(118°) ≈ 0.882948 ≈ 0.88

Putting these values into the formula gives ...

  Area = (1/2)(5.9 cm)(7.2 cm)(0.88) = 18.6912 cm² ≈ 18.7 cm² . . . rounded

You will observe that this answer does not match any offered choice.

__

Rounding only at the End

The preferred method of working these problems is to keep the full precision the calculator offers until the final answer is achieved. Then appropriate rounding is applied. Using this solution method, we get ...

  Area = (1/2)(5.9 cm)(7.2 cm)(0.882948) ≈ 18.7538 cm² ≈ 18.8 cm²

This answer matches the first choice.

__

Using the 3 Side Lengths

Since the figure includes all three side lengths, we can compute a more precise value for angle C, or we can use Heron's formula for the area of the triangle. Each of these methods will give the same result.

From the Law of Cosines, the angle C is ...

  C = arccos((a² +b² -c²)/(2ab)) = arccos(-38.79/84.96) ≈ 117.16585°

Note that this is almost 1 full degree less than the angle shown in the diagram. Then the area is ...

   Area = (1/2)(5.9 cm)(7.2 cm)sin(117.16585°) ≈ 18.8970 cm² ≈ 18.9 cm²

This answer may be the most accurate yet, but does not match any offered choice.

Answer:

D.

Step-by-step explanation:

That would be D because there is a repetition of x = -6.

-6 maps to -6, 5 and -5 which is not allowed in a function.

Functions can be one-to-one or many-to-one but not one-to-many.

Answer:

Option A (Exterior)

Step-by-step explanation:

To understand this question, it is important to understand the concept of the exterior angle. An exterior angle is an angle which is made by two intersecting lines outside of the shape. Basically, one of the two lines is extended outside the shape. The angle between the extended line and the other line which is not extended is the exterior angle. It is outside the shape. The interior angle is the angle which is made by the same two lines but inside the shape.

The sum of the interior angle and the exterior angle is 180 degrees. It is also interesting to note that the sum of the angles in the triangle is 180 degrees.

Suppose that the angles in the triangles are A, B, and C, and the associated exterior angle with the angle A is angle D. By the argument, A+B+C=180 degrees and A+D=180 degrees. Since 180 degrees = 180 degrees, therefore A+B+C = A+D. Angle A cancels on both sides and reduces to B+C=D. This proves that the exterior angle of a triangle is equal to the sum of the two remote interior angles!!!

Answer: We can expect about 40.13% of bottles to have a volume less than 32 oz.

Step-by-step explanation:

Given : The volumes of soda in quart soda bottles can be described by a Normal model with

[tex]\mu=\text{32.3 oz}\\\\\sigma=\text{1.2 oz}[/tex]

Let X be the random variable that represents the volume of a randomly selected bottle.

z-score :[tex]\dfrac{x-\mu}{\sigma}[/tex]

For x = 32 oz

[tex]z=\dfrac{32-32.3}{1.2}=-0.25[/tex]

The probability of bottles have a volume less than 32 oz is given by :-

[tex]P(X<32)=P(z<-0.25)=0.4012937[/tex]           [Using standard normal table]

In percent, [tex]0.4012937\times100=40.12937\%\approx40.13\%[/tex]

Hence, we can expect about 40.13% of bottles to have a volume less than 32 oz.

Answer:

Step-by-step explanation:

The given side is opposite the given acute angle in this right triangle, so the applicable relation is ...

  Sin(25°) = CB/AB

Solving for AB, we get ...

  AB = CB/sin(25°) ≈ 37.859

__

The relation involving the other leg of the triangle is ...

  Tan(25°) = CB/AC

Solving for AC, we get ...

  AC = CB/tan(25°) ≈ 34.312

__

Of course, the missing angle is the complement of angle A, so is 90-25 = 65 degrees.

Hence, the average rate of change in vertical height is:

                               -6

We know that the average amount that the roller coaster's height changes over each horizontal foot is basically the slope or the average rate of change of the height of the roller coaster to the horizontal distance.

i.e. it is the ratio of the vertical change i.e. the change in height of the roller coaster to the horizontal change.

Here the vertical change= -150 feet

and horizontal change = 25 feet

Hence,

Average rate of change is:

[tex]=\dfrac{-150}{25}\\\\=-6[/tex]

So, for every change in horizontal distance by 1 feet the vertical height drop by 6 feet.

Answer:

The average amount that the roller coaster's height changes over each horizontal foot is -6.

Further explanation:

The rate of linear function is known as the slope. And the slope can be defined as the ratio of vertical change (change in y) to the horizontal change (change in x).

Mathematically, we can write

[tex]\text{Slope}=\dfrac{\text{change in y}}{\text{change in x}}=\dfrac{\Delta y}{\Delta x}[/tex]

Now, we have been given that  

Roller coaster has a steep drop at a horizontal distance of 25 feet.

Thus, [tex]\Delta x=25\text{ feet}[/tex]

The height of the roller coaster at the bottom of the drop is -150 feet.

Thus, [tex]\Delta y=-150\text{ feet}[/tex]

Using the above- mentioned formula, the average rate of change is given by

[tex]\text{Average rate of change }=\dfrac{-150}{25}[/tex]

On simplifying the fraction

[tex]\text{Average rate of change }=\dfrac{-6}{1}=-6[/tex]

It means for every 1 foot of horizontal distance, the roller coaster moves down by 6 feet.  

Please refer the attached graph to understand it better.

Therefore, we can conclude that the average amount that the roller coaster's height changes over each horizontal foot is -6.

Learn more:

Average rate of change: https://brainly.com/question/10961592

Finding Average: https://brainly.com/question/9145375

Keywords:

Average rate of change, slope, change of y over change of x, the ratio of two numbers be the same.

Answer:

The correct option for the provided problem is D. A selection or listing of objects in which the order of the objects is not important.

Step-by-step explanation:

Consider the provided information.

Selecting all the parts of a set of objects without considering its order in which the objects are selecting is known as combination.

Now consider the provided options:

Options A, B, and C are not valid as the description does not fits accurately.

Thus, the correct option for the provided problem is D. A selection or listing of objects in which the order of the objects is not important.

Answer:

A selection or listing of objects in which the order of the objects is not important

Step-by-step explanation:

Answer: 5 units

Step-by-step explanation:

The formula to find the area of a triangle is given by :-

[tex]\text{Area}=\dfrac{1}{2}\text{ base * height}[/tex]

Given : The area of a triangle = 30 square units

The length of the base of the triangle = 12 units

Let h be the height of the triangle .

Then , we have

[tex]30=\dfrac{1}{2}12\times h\\\\\Rightarrow\ h=\dfrac{30}{6}\\\\\Rightarrow\ h=5\text{ units}[/tex]

Hence, the height of a triangle = 5 units

Answer:

(x − 3)² + (y − 4)² = 5

Step-by-step explanation:

The equation of a circle is:

(x − h)² + (y − k)² = r²

where (h, k) is the center and r is the radius.

First use the distance formula to find the radius:

d² = (x₂ − x₁)² + (y₂ − y₁)²

r² = (2 − 3)² + (6 − 4)²

r² = 1 + 4

r² = 5

Given that (h, k) = (3, 4):

(x − 3)² + (y − 4)² = 5

Answer:

Step-by-step explanation:

Inserting the coordinates of the center (3, 4) into the standard equation of a circle with center at (h, k) and radius r, we get:

(x - 3)^2 + (y - 4)^2 = r^2

Next, we substitute 2 for x, 6 for y and solve the resulting equation for r^2:

(2 - 3)^2 + (6 - 4)^2 = r^2, or

      1      +       4       =  r^2.

Thus, the radius is √5.  Subbing this result into the equation found above, (x - 3)^2 + (y - 4)^2 = r^2, we get:

(x - 3)^2 + (y - 4)^2 = (√5)^2 = 5, which matches the last of the four possible answer choices.

Answer:

The vertex is the point (-6,-34)

Step-by-step explanation:

we know that

The equation of a vertical parabola into vertex form is equal to

[tex]y=a(x-h)^{2}+k[/tex]

where

(h,k) is the vertex of the parabola

In this problem we have

[tex]y=x^{2}+12x+2[/tex]

Convert in vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex]y-2=x^{2}+12x[/tex]

Complete the square . Remember to balance the equation by adding the same constants to each side.

[tex]y-2+36=x^{2}+12x+36[/tex]

[tex]y+34=x^{2}+12x+36[/tex]

Rewrite as perfect squares

[tex]y+34=(x+6)^{2}[/tex]

[tex]y=(x+6)^{2}-34[/tex]

The vertex is the point (-6,-34)

Answer:

The measure of arc BC = 124°

Step-by-step explanation:

From the figure we can write,

measure of arc AB + measure of arc BC + measure of arc AC = 360

measure of arc AB   = 146°

measure of arc BC = 90°

Therefore measure arc BC = 360 - (146 + 90)

 = 360 - 236

 = 124°

The measure of arc BC = 124°

Answer: 124 degrees

Step-by-step explanation: There is a 90 degree angle in the top right of the circle. There is a 146 degree angle. Add these two angles.

90 + 146 = 236

These two angles combined are 236 degrees. We are trying to find BC, which is the rest of the circle. There are 360 degrees in a circle. Subtract 360 from 236.

360 - 236 = 124

BC = 124 degrees.

Answer:

D x=1

Step-by-step explanation:

6^(3-x)=6^2

Since the bases are the same, the exponents have to be the same

3-x = 2

Subtract 3 from each side

3-x-3 = 2-3

-x = -1

Multiply each side by -1

x = 1

Answer: Option D

[tex]x=1[/tex]

Step-by-step explanation:

We have the following exponential equation

[tex]6^{3-x}=6^2[/tex]

We must solve the equation for the variable x

Note that the exponential expressions [tex]6^{3-x}[/tex] and [tex]6 ^ 2[/tex] have the same base: 6

So if [tex]6^{3-x}=6^2[/tex] this means that [tex]3-x = 2[/tex]

Then we have that:

[tex]3-x = 2[/tex]

[tex]x = 3-2\\x=1[/tex]

Answer:

  3

Step-by-step explanation:

The diagram shows the answer: 3 pictures will fit vertically.

You can solve this algebraically as well. For n pictures, there will be n-1 spaces, so the total height of the page must satisfy ...

  1 + 3n + 1/2(n -1) + 1 ≤ 12

  3.5n + 1.5 ≤ 12 . . . . . . . . . . . simplify

  3.5n ≤ 10.5 . . . . . . . . . . . . . .subtract 1.5

  n ≤ 3 . . . . . . . . . . . . . . . . . . . divide by 3.5

Up to 3 pictures will fit in a column.

Answer:

True

Step-by-step explanation:

Segments drawn to a circle from the same outside point are congruent.

Segments YZ and XY are tangent to circle E draw from outside point Y. The segments are congruent, so the statement is true.

Answer:

Option B After 14 years the car is worth $0

Step-by-step explanation:

we have

[tex]V=-1,385n+18,790[/tex]

where

V is the value of the cars

n is the number of years

Determine the n-intercept of the graph

we know that

The n-intercept is the value of n (number of years) when the value of V (value of the car) is equal to 0

so

For V=0

substitute and solve for n

[tex]0=-1,385n+18,790[/tex]

[tex]1,385n=18,790[/tex]

[tex]n=18,790/1,385[/tex]

[tex]n=14\ years[/tex]

That means

After 14 years the car is worth $0

Answer:

B

Step-by-step explanation:

Answer: Option C

No; Jamie could not have won 11 games because [tex]2x - 5 \neq 30[/tex]

Step-by-step explanation:

Let's call x the number of games that Jamie has won

Let's call y the number of games that Imani has won

We know that Imani has won 5 more games than Jamie.

Then we can say that:

[tex]y= x - 5[/tex]

We know that the total number of games that Jamie and Imani have won together is 30.

So

[tex]x + y = 30[/tex]

We want to know if it is possible that [tex]x = 11[/tex].

Then we substitute the first equation in the second and get the following:

[tex]x + x - 5 =30\\2x - 5 = 30[/tex]

Now replace [tex]x = 11[/tex] in the equation and check if equality is met.

[tex]2 (11) - 5 = 30\\22 - 5 = 30\\17 \neq 30[/tex]

Equality is not met, then the correct answer is option C

Answer: is c (no Jamie could not have won 11 games because 2x-5=/30

Step-by-step explanation:

Answer:

P: 20pi A: 400-100pi

Answer:

  291.9518 T are required for completion

Step-by-step explanation:

The remaining 0.62 part is ...

  0.62 × 470.89 T = 291.9518 T

Answer:

291.9518 metric Ton

Step-by-step explanation:

Hello

according to the data provided by the problem.

Total Iron needed to build the railing (A)= 470.89 Ton

Total Iron purchased by the factory =0.38 of total

Total Iron purchased by the factory =0.38 *470.89

Total Iron purchased by the factory (B)=178.9382metric Ton

the difference between the total iron needed and the iron supplied by the factory will be the iron we need to get

A-B=iron we need to get(c)

C=A-B

C=470.89-178.9382

C=291.9518 metric Ton

Have a great day.

Answer:

t = 21.97 years

Step-by-step explanation:

The formula for the continuous compounding if given by:

A = p*e^(rt); where A is the amount after compounding, p is the principle, e is the mathematical constant (2.718281), r is the rate of interest, and t is the time in years.

It is given that p = $x, r = 5%, and A = $3x. In this part, t is unknown. Therefore: 3x = x*e^(0.05t). This implies 3 = e^(0.05t). Taking natural logarithm on both sides yields ln(3) = ln(e^(0.05t)). A logarithmic property is that the power of the logarithmic expression can be shifted on the left side of the whole expression, thus multiplying it with the expression. Therefore, ln(3) = 0.05t*ln(e). Since ln(e) = 1, and making t the subject gives t = ln(3)/0.05. This means that t = 21.97 years (rounded to the nearest 2 decimal places)!!!

Answer:

t = 22 years

Step-by-step explanation:

* Lets explain the compound continuous interest

- Compound continuous interest can be calculated using the formula:

  A = P e^rt

# A = the future value of the investment, including interest

# P = the principal investment amount (the initial amount)

# r = the interest rate  

# t = the time the money is invested for

- The formula gives you the future value of an investment,  

  which is compound continuous interest plus the

  principal.  

* Now lets solve the problem

∵ The initial investment amount is P

∵ The future amount after t years is three times the initial value

∴ A = 3P

∵ The rate of interest is 5%

∴ r = 5/100 = 0.05

- Lets use the rule above to find t

∵ A = P e^rt

∴ 3P = P e^(0.05t)

- Divide both sides by P

∴ 3 = e^(0.05t)

- Insert ㏑ for both sides

∴ ㏑(3) = ㏑(e^0.05t)

- Remember ㏑(e^n) = n ㏑(e) and ㏑(e) = 1, then ㏑(e^n) = n

∴ ㏑(3) = 0.05t

- Divide both sides by 0.05

∴ t = ㏑(3)/0.05 = 21.97 ≅ 22

* t = 22 years

Answer:

Proofs are in the explanation.

Step-by-step explanation:

b)  My first thought is to divide top and bottom on the left hand side by [tex]\cos(\alpha)[/tex].

I see this would give me 1 on top and where that sine is, it would give me tangent since sine/cosine=tangent.

Let's do it and see:

[tex]\frac{\cos(\alpha)}{\cos(\alpha)-\sin(\alpha)} \cdot \frac{\frac{1}{\cos(\alpha)}}{\frac{1}{\cos(\alpha)}}[/tex]

[tex]=\frac{\frac{\cos(\alpha)}{\cos(\alpha)}}{\frac{\cos(\alpha)}{\cos(\alpha)}-\frac{\sin(\alpha)}{\cos(\alpha)}}[/tex]

[tex]=\frac{1}{1-\tan(\alpha)}[/tex]

c) My first idea here is to expand the cos(x+y) using the sum identity for cosine.

So let's do that:

[tex]\frac{\cos(x)\cos(y)-\sin(x)\sin(y)}{\cos(x)\sin(y)}[/tex]

Separating the fraction:

[tex]\frac{\cos(x)\cos(y)}{\cos(x)\sin(y)}-\frac{\sin(x)\sin(y)}{\cos(x)\sin(y)}[/tex]

The cos(x) cancel's in the first fraction and the sin(y) cancels in the second fraction:

[tex]\frac{\cos(y)}{\sin(y)}-\frac{\sin(x)}{\cos(x)}[/tex]

[tex]\cot(y)-\tan(x)[/tex]

d) This one makes me think it is definitely essential that we use properties of logarithms.

The left hand side can be condense into one logarithm using the product law:

[tex]\ln|(1+\cos(\theta))(1-\cos(\theta))|[/tex]

We are multiplying conjugates inside that natural log so we only need to multiply the first and the last:

[tex]\ln|1-\cos^2(\theta)|[/tex]

I can rewrite [tex]1-\cos^2(\theta)[/tex] using the Pythagorean Identity:

[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]:

[tex]\ln|\sin^2(\theta)|[/tex]

Now by power rule for logarithms:

[tex]2\ln|\sin(\theta)|[/tex]

Answer:  [tex]\dfrac{20}{3}\text{ hours}[/tex]

Step-by-step explanation:

Let x be the speed of slower pump and 1.5x be the speed of faster pump to fill the swimming pool .

Then , According to the given question, we have the following equation:-

[tex]x+1.5x=\dfrac{1}{4}\\\\\rightarrow\ 2.5x=\dfrac{1}{4}\\\\\Rightarrow\ x=\dfrac{1}{10}=[/tex]

Now, the time taken by faster pump to fill the pool is given by :-

[tex]t=\dfrac{1}{1.5x}=\dfrac{10}{1.5}=\dfrac{20}{3}\text{ hours}[/tex]

Hence, the faster pump would take [tex]\dfrac{20}{3}\text{ hours}[/tex]  to fill the pool if it had worked alone at its constant rate.

Answer:

  4 m

Step-by-step explanation:

Use the formula for the area of a triangle. Fill in the given numbers and solve for the unknown.

  A = (1/2)bh

  48 m² = (1/2)(24 m)h . . . . . put in the given numbers

  (48 m²)/(12 m) = h = 4 m . . . . divide by the coefficient of h

The height of the triangle is 4 meters.

Answer with explanation:

Pre-image =Rectangle EFGH

Image = Rectangle E'F'G'H'

Stretch Factor = 2.5

Coordinates of Point H= (-2,0)

If Coordinate of any point is (x,y) and it is stretched by a factor of k , then coordinate of that point after stretching = (k x , k y).

So, Coordinates of Point H' will be=(-2×2.5,0×2.5)

                                      = (-5,0)

Answer: (-5,0)

Step-by-step explanation:

Given : Square EFGH stretches vertically by a factor of 2.5 to create rectangle  E?F?G?H?.

The square stretches with respect to the x-axis such that the point H is located at (-2, 0).

Since , we know that to find the coordinate of image , we multiply the scale factor to the coordinate of pre-image.

Then , the coordinate of H? is given by :-

[tex](-2\times2.5, 0\times2.5)=(-5,0)[/tex]