Use the graphs shown in the figure below. All have the form f(x) = abª. Which graph has the smallest value for b? ​

Answer: The answer is D

Step-by-step explanation:

Pythagorean theorem: a²+b²=c²

x²+x²=14²

2x²=196

Evaluate...

x=7√2

Since Kingsley wanted an equation to convert from inches to centimeters, the correct answer is B) c = 2.54i.

What is equation ?

An equation is a statement that asserts the equality of two expressions, usually separated by an equals sign (=). The expressions on either side of the equals sign may contain one or more variables, which are unknown values that can be determined by solving the equation.

Kingsley wants to convert a given length in inches to centimeters. He knows that 1 inch is about 2.45 centimeters.

Let's call the length in inches "i" and the length in centimeters "c".

We want to find an equation that relates i and c. We know that 1 inch is about 2.45 centimeters, so we can write:

1 inch = 2.45 centimeters

To convert from inches to centimeters, we can multiply the length in inches by 2.45. So:

c = 2.45i

This is the equation Kingsley can use to convert any given length in inches to centimeters.

Alternatively, we can rearrange this equation to solve for i:

c = 2.45i

Divide both sides by 2.45:

c/2.45 = i

So the equation for converting from centimeters to inches is:

i = c/2.45

Therefore,  since Kingsley wanted an equation to convert from inches to centimeters, the correct answer is B) c = 2.54i.

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Step-by-step explanation:

There are many possible solutions to this problem, but one possible set of values for P, Q, R, S, T, and U is:

P = 2

Q = 5

R = 1

S = 8

T = 9

U = 9

With these values, we have:

PQR = 251

STU = 156

And the sum of PQR and STU is indeed 407.

Answer:

Find f(g(x)) f(x)=7x-8 , g(x)=3x-2. f(x)=7x−8 f ( x ) = 7 x - 8 , g(x)=3x−2 g ( x ) = 3 x - 2. Step 1. Set up the composite result function. f(g(x)) f ( g ...

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Step 1: Arrange the arrays so that A and B are in the same order: A = [ 1 2 4 0 5 6 ], B = [ 7 3 2 5 1 9]

Step 2: To find C = A+B, add each element of A and B together.

C = [1+7, 2+3, 4+2, 0+5, 5+1, 6+9]

C = [8, 5, 6, 5, 6, 15]

Step 3: To find D = A-B, subtract each element of B from A.

D = [1-7, 2-3, 4-2, 0-5, 5-1, 6-9]

D = [-6, -1, 2, -5, 4, -3]

The actual intake of carbohydrates is 138% as compare to recommended intake.

Recommended intake of carbohydrates or any other nutrient are,

Based on the information provided,

Consumed 159 grams of carbohydrates,

Recommended intake is between 115 and 166 grams.

Calculate the percentage of actual intake compared to the recommended intake, use the following formula,

Percentage = (Actual Intake / Recommended Intake) x 100%

Substituting the values in the formula we have,

⇒Percentage = (159 / 115) x 100%

⇒Percentage ≈ 138.3%

Therefore,  the actual intake of carbohydrates is about 138% of the recommended intake, indicating that consumption of more carbohydrates than recommended.

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Answer: numbers 1,3 and 4

Step-by-step explanation:

Answer:

One possible function that models the ocean tide is:

h(t) = A sin(ωt + φ) + B

where:

h(t) represents the height of the tide (in meters) at time t (in hours)

A is the amplitude of the tide (in meters)

ω is the angular frequency of the tide (in radians per hour)

φ is the phase shift of the tide (in radians)

B is the mean sea level (in meters)

This function is a sinusoidal function, which is a common type of function used to model periodic phenomena. The sine function has a natural connection to circles and periodic motion, making it a good choice for modeling the regular rise and fall of ocean tides.

The amplitude A represents the maximum height of the tide above the mean sea level, while B represents the mean sea level. The angular frequency ω determines the rate at which the tide oscillates, with one full cycle (i.e., a high tide and a low tide) occurring every 12 hours. The phase shift φ determines the starting point of the tide cycle, with a value of zero indicating that the tide is at its highest point at time t=0.

To determine specific values for A, ω, φ, and B, we would need to gather data on the tide height at various times and locations. However, typical values for these parameters might be:

1. A = 2 meters (representing a relatively large tidal range)

2. ω = π/6 radians per hour (corresponding to a 12-hour period)

3. φ = 0 radians (assuming that high tide occurs at t=0)

4. B = 0 meters (assuming a mean sea level of zero)

Using these values, we can write the equation for the tide as:

h(t) = 2 sin(π/6 t)

We can evaluate this equation for various values of t to get the height of the tide at different times. For example, at t=0 (the start of the cycle), we have:

h(0) = 2 sin(0) = 0

indicating that the tide is at its lowest point. At t=6 (halfway through the cycle), we have:

h(6) = 2 sin(π/2) = 2

indicating that the tide is at its highest point. We can also graph the function to visualize the rise and fall of the tide over time:

Tide Graph

Overall, this function provides a simple and effective way to model the ocean tide using trigonometric functions.

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Answer:

The value of the derivative at (-2/3, 2√3/3) is zero.

Step-by-step explanation:

Given function:

[tex]f(x)=-3x\sqrt{x+1}[/tex]

To differentiate the given function, use the product rule and the chain rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Product Rule of Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{7 cm}\underline{Differentiating $[f(x)]^n$}\\\\If $y=[f(x)]^n$, then $\dfrac{\text{d}y}{\text{d}x}=n[f(x)]^{n-1} f'(x)$\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Let}\;u &= -3x& \implies \dfrac{\text{d}u}{\text{d}{x}} &= -3\\\\\textsf{Let}\;v &= \sqrt{x+1}& \implies \dfrac{\text{d}v}{\text{d}{x}} &=\dfrac{1}{2} \cdot (x+1)^{-\frac{1}{2}}\cdot 1=\dfrac{1}{2\sqrt{x+1}}\end{aligned}[/tex]

Apply the product rule:

[tex]\implies f'(x) =u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}[/tex]

[tex]\implies f'(x)=-3x \cdot \dfrac{1}{2\sqrt{x+1}}+\sqrt{x+1}\cdot -3[/tex]

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-3\sqrt{x+1}[/tex]

Simplify:

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-\dfrac{3\sqrt{x+1} \cdot 2\sqrt{x+1}}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{3x}{2\sqrt{x+1}}-\dfrac{6(x+1)}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{3x+6(x+1)}{2\sqrt{x+1}}[/tex]

[tex]\implies f'(x)=- \dfrac{9x+6}{2\sqrt{x+1}}[/tex]

An extremum is a point where a function has a maximum or minimum value.

From inspection of the given graph, the maximum point of the function is (-2/3, 2√3/3).

To determine the value of the derivative at the maximum point, substitute x = -2/3 into the differentiated function.

[tex]\begin{aligned}\implies f'\left(-\dfrac{2}{3}\right)&=- \dfrac{9\left(-\dfrac{2}{3}\right)+6}{2\sqrt{\left(-\dfrac{2}{3}\right)+1}}\\\\&=-\dfrac{0}{2\sqrt{\dfrac{1}{3}}}\\\\&=0 \end{aligned}[/tex]

Therefore, the value of the derivative at (-2/3, 2√3/3) is zero.

Answer:

Let's assume that Mr. Nkalle invested an amount of x in Scheme A and (20900 - x) in Scheme B.

The simple interest earned on Scheme A in 2 years would be:

SI(A) = (x * 9 * 2)/100 = 0.18x

The simple interest earned on Scheme B in 2 years would be:

SI(B) = [(20900 - x) * 8 * 2]/100 = (3344 - 0.16x)

The total simple interest earned in 2 years is given as N$3508:

SI(A) + SI(B) = 0.18x + (3344 - 0.16x) = 3508

0.02x = 164

x = 8200

Therefore, Mr. Nkalle invested N$8200 in Scheme A and N$12700 (20900 - 8200) in Scheme B. So the amount invested in Scheme B was N$12700.

Yes, there is an analogous ratio between 2:1 and 20:10.

We just cancel by a common factor. So 4:2=2:1 . The simplest representation of the ratio 4 to 2 is the ratio 2 to 1. Also, since each pair of numbers has the same relationship to one another, the ratios are equivalent.

By dividing the terms of each ratio by their greatest common factor, we may simplify both ratios to explain why.

As the greatest common factor for the ratio 2:1 is 1, additional simplification is not necessary.

The greatest common factor for the ratio 20:10 is 10. When we multiply both terms by 10, we get:

20 ÷ 10 : 10 ÷ 10

= 2 : 1

As a result, both ratios have the same reduced form, 2:1, making them equal.

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The direction angle of the electric field at the point (x = 1.10 m, y = 0.400 m, z = 0) is approximately 74.5 degrees clockwise from the positive x-axis in the xy plane.

To calculate the electric field at the point (x = 1.10 m, y = 0.400 m, z = 0), we need to take the negative gradient of the electric potential V:

E = -∇V

where ∇ is the del operator, which is given by:

∇ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z)

and i, j, k are the unit vectors in the x, y, and z directions, respectively.

To calculate the magnitude of the electric field at the point, we first need to find the partial derivatives of V with respect to x and y:

∂V/∂x = 2Axy - By^2

∂V/∂y = Ax^2 - 2Bxy

Substituting the values of A, B, x, and y, we get:

∂V/∂x = 2(5.00 V/m^3)(1.10 m)(0.400 m) - (8.00 V/m^3)(0.400 m)^2 = 0.44 V/m

∂V/∂y = (5.00 V/m^3)(1.10 m)^2 - 2(8.00 V/m^3)(1.10 m)(0.400 m) = -1.64 V/m

Next, we can calculate the magnitude of the electric field:

E = -∇V = -i(∂V/∂x) - j(∂V/∂y) - k(∂V/∂z)

= -i(0.44 V/m) + j(1.64 V/m) + 0k

= (0.44 i - 1.64 j) V/m

The magnitude of the electric field is given by:

|E| = sqrt((0.44 V/m)^2 + (-1.64 V/m)^2) = 1.70 V/m

Therefore, the magnitude of the electric field at the point (x = 1.10 m, y = 0.400 m, z = 0) is 1.70 V/m.

To calculate the direction angle of the electric field, we need to find the angle that the electric field vector makes with the positive x-axis in the xy plane.

The angle can be found using the arctan function:

θ = arctan(Ey/Ex)

Substituting the values of Ex and Ey, we get:

θ = arctan(-1.64 V/m / 0.44 V/m) = -1.30 radians

The negative sign indicates that the direction angle is measured counter clockwise from the negative x-axis, which is equivalent to measuring clockwise from the positive x-axis.

Converting to degrees, we get:

θ = -1.30 radians * (180 degrees / pi radians) = -74.5 degrees

Therefore, the direction angle is approximately 74.5 degrees clockwise in the xy plane.

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According to the question. A. No punctuation is missing.

Punctuation is the use of symbols to indicate the structure and organization of written language. It is used to help make the meaning of sentences clearer and to make them easier to read and understand. Punctuation marks can also be used to indicate pauses in speech, to create emphasis, and to indicate the speaker’s attitude. There are many different types of punctuation marks, each with its own purpose. The most commonly used punctuation marks are the period, comma, question mark, exclamation mark, quotation marks, and the apostrophe.

Quotation marks are used to enclose quoted material, while the apostrophe is used to indicate possession or to replace missing letters in a word or phrase. By using punctuation correctly, writers can ensure that their messages are correctly understood by their readers.

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Answer:

y = 4 sin(π x) + 5

Step-by-step explanation:

A sine function with a midline of y=5, an amplitude of 4, and a period of 2 can be written in the following form:

y = A sin(2π/ x) +

where A is the amplitude, is the period, is the vertical shift (midline), and x is the independent variable (usually time).

Substituting the given values, we get:

y = 4 sin(2π/2 x) + 5

Simplifying this expression, we get:

y = 4 sin(π x) + 5

Therefore, the sine function with the desired characteristics is:

y = 4 sin(π x) + 5

The average rate of change on the intervals [0, 3], [3, 5], [5, 7], and [7, 9] are 2, -1.5, 1, and -1.5, respectively.

It expresses how much the function changed per unit on average during that time period. It is computed by taking the slope of the straight line connecting the interval's endpoints on the function's graph.

To calculate the average rate of change for the intervals shown in the graph, we must first determine the slope of the line connecting the endpoints of each interval.

0-3 interval:

Because the interval's endpoints are (0, 1) and (3, 7), the slope of the line connecting them is:

slope = (y change) / (x change) = (7 - 1) / (3 - 0) = 2

pauses [3, 5]:

Because the interval's endpoints are (3, 7) and (5, 4), the slope of the line connecting them is:

slope = (y change) / (x change) = (4 - 7) / (5 - 3) = -1.5

[5–7] Interval:

Because the interval's endpoints are (5, 4) and (7, 6), the slope of the line connecting them is:

slope = (y change) / (x change) = (6 - 4) / (7 - 5) = 1

Interval 7 and 9:

Because the interval's endpoints are (7, 6) and (9, 3), the slope of the line connecting them is:

slope = (y change) / (x change) = (3 - 6) / (9 - 7) = -1.5

As a result, the average rate of change on the intervals [0, 3], [3, 5], [5, 7], and [7, 9] is 2, -1.5, 1, and -1.5.

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The given statement "Star clusters with lots of bright, blue stars of spectral type O and B are generally younger than clusters that don't have any such stars." is True. The reason for this is that O and B stars are short-lived and burn through their fuel quickly.

The reason for this is that O and B stars burn through their fuel quickly, causing them to exhaust their nuclear fuel and end their lives in a relatively short period, typically within a few tens of millions of years.

On the other hand, stars of lower mass and cooler temperatures, like G and K type stars like our sun, have longer lifetimes and take billions of years to exhaust their nuclear fuel.

Therefore, clusters without any bright, blue stars are likely to have evolved for longer periods, allowing these short-lived stars to have already expired.

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Answer:

The answer is brand B

Step-by-step explanation:

You divide $14.88 by 24 which equals 68 cents per item.

Then brand B is 60 cents per item which is the better buy!

The value of h(x) using exponents are as follows:

For -1, the value of h(x)=1/10

For 0, the value of h(x) = 1

For 1, the value of h(x) = 10

For 2, the value of h(x) = 100

For 3, the value of h(x) = 1000

The exponent of a number tells us how many times the original value has been multiplied by itself. For instance, 2×2×2×2 can be expressed as [tex]2^{4}[/tex] the result of 4 times multiplying 2 by itself. Thus, 4 is referred to as the "exponent" or "power," while 2 is referred to as the "base."

Generally speaking, [tex]x^{n}[/tex] denotes that x has been multiplied by itself n times. Here x is the base and n is the power.

Now here, as we put the value of x in the equation, h(x) we can get the value of h(x) for each value of x.

So,

For -1, the value of h(x)=1/10

For 0, the value of h(x) = 1

For 1, the value of h(x) = 10

For 2, the value of h(x) = 100

For 3, the value of h(x) = 1000

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Days when change in temperature more than 10° F are Option B)Tuesday and E) Friday.

calculating the difference by deducting the end temperature from the initial temperature. The temperature difference is therefore 75 degrees Celsius - 50 degrees Celsius = 25 if something begins at 50 degrees Celsius and ends at 75 degrees Celsius.

Change in temperature on Monday from High to low

=15-10=5°F

Change in temperature on Tuesday from High to low

=8-(-4)=12°F

Change in temperature on Wednesday from High to low

=-2-(-5)=3°F

Change in temperature on Thursday from High to low

=-3-(-7)=4°F

Change in temperature on Friday from High to low

=-1-(-12)=11° F

Days when change in temperature more than 10° F are Tuesday and Friday.

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The Complete question is attached below:

Answer: D. Square

Step-by-step explanation:

The shape created by the cross section of the cut through a square pyramid is a square.

To see why, imagine the pyramid sitting on a table with its square base flat against the surface. The cut goes through the vertex, which is the point at the top of the pyramid. Since the cut is perpendicular to the base, it divides the pyramid into two smaller pyramids with congruent, but not identical, bases. Each of these smaller pyramids has a triangular base that is an isosceles right triangle. The two triangles share a common hypotenuse, which is the line of the cut.

The cross section of the cut is the shape formed where the two triangles meet along the hypotenuse. Since both triangles are congruent and the hypotenuse is the same for both, the cross section is a square. The sides of the square are equal to the base of the original pyramid, which is one of the legs of the isosceles right triangles formed by the cut. Therefore, the answer is D, a square.

On the 7th day after the videο was pοsted, 640 peοple will see the videο.

Statistics is the discipline that cοncerns the cοllectiοn, οrganizatiοn, analysis, interpretatiοn, and presentatiοn οf data.

1 Let's start by finding the pattern in the number οf viewers. We knοw that οn the first day, 5 peοple watched the videο. On the next day, the number οf viewers dοubled tο 5 x 2 = 10. On the third day, the number οf viewers dοubled again tο 10 x 2 = 20. We can see that the number οf viewers is dοubling each day, which means we can write the number οf viewers as:

[tex]V = 5 x 2^n[/tex]

where n is the number οf days after the videο was pοsted.

Nοw we want tο find οn which day the number οf viewers will be 640. Sο we can set V equal tο 640 and sοlve fοr n:

[tex]640 = 5 x 2^n[/tex]

[tex]2^n = 128[/tex]

n = lοg2(128) = 7

2.   Tο find the first day when mοre than 20,000 peοple will see the videο, we can set V equal tο 20,000 and sοlve fοr n:

[tex]20,000 = 5 x 2^n[/tex]

2^n = 4,000

n = lοg2(4,000) ≈ 11.29

Since n represents the number οf days after the videο was pοsted, we can rοund up tο the next whοle number tο find the first day when mοre than 20,000 peοple will see the videο. Therefοre, οn the 12th day after the videο was pοsted, mοre than 20,000 peοple will see the videο if the trend cοntinues

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Uri paid a total of $85.47 for the landscaping service including tax and tip.

Taxes are compulsory payments made by a government organisation, whether local, regional, or federal, to people or businesses. Tax revenues are used to fund a variety of government initiatives, such as Social Security and Medicare as well as public infrastructure and services like roads and schools. Taxes are borne by whoever bears the cost of the tax in economics, whether this is the entity being taxed, such as a business, or the final users of the items produced by the firm. Taxes should be taken into consideration from an accounting standpoint, including payroll taxes, federal and state income taxes, and sales taxes.

Given that company charged $74 for the service plus 5% tax.

The tax is 5%, that is:

Tax = 5% of $74 = 0.05 x $74 = $3.70

Cost after tax = $74 + $3.70 = $77.70

Now, tip is 10%:

Tip = 10% of $77.70 = 0.10 x $77.70 = $7.77

Total cost = $77.70 + $7.77 = $85.47

Hence, Uri paid a total of $85.47 for the landscaping service including tax and tip.

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The function represented by the table is linear, as it has a constant rate of change and is represented by a straight line.

Function in mathematics is a relation between two sets, where one set is the input and the other set is the output. Functions are an important tool in mathematics and can be used to describe and model real-world phenomena. Functions take inputs, manipulate them and produce outputs. They can be used to represent relationships between two or more variables, or to represent a complex process. Functions allow us to break down complex problems into smaller, more manageable pieces and to study how changes in one variable affect other variables.

The function represented by the table is linear. It can be determined by the fact that the y-values change by the same amount every time the x-values increase by one unit. In this case, the y-values decrease by 2 each time the x-values increase by one unit. This is an example of a linear function.

Linear functions have the shape of a straight line and are characterized by having a constant rate of change. The constant rate of change is represented by the slope of the line, which in this case is -2. This means that for every one unit increase in the x-values, the y-values decrease by two.

A quadratic function is the opposite of a linear function, as it has a rate of change that is not constant. Quadratic functions are characterized by their parabolic shape and their rate of change increases as x-values increase. Exponential functions are characterized by their curved shape and increase exponentially as x-values increase.

In conclusion, the function represented by the table is linear, as it has a constant rate of change and is represented by a straight line.

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4775 g968r648 747474874 483892874 23773259635y84b2375789325 7437594365825 4378574937587 49388959365n 98437858746587 32o4iy548569

Answer:

?

Step-by-step explanation:

[tex]\begin{cases} x=\frac{2+i\sqrt{2}}{3}\implies 3x=2+i\sqrt{2}\implies 3x-2-i\sqrt{2}=0\\\\ x=\frac{2-i\sqrt{2}}{3}\implies 3x=2-i\sqrt{2}\implies 3x-2+i\sqrt{2}=0 \end{cases} \\\\\\ \stackrel{ \textit{original polynomial} }{a(3x-2-i\sqrt{2})(3x-2+i\sqrt{2})=\stackrel{ 0 }{y}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{ \textit{difference of squares} }{[(3x-2)-(i\sqrt{2})][(3x-2)+(i\sqrt{2})]}\implies (3x-2)^2-(i\sqrt{2})^2 \\\\\\ (9x^2-12x+4)-(2i^2)\implies 9x^2-12x+4-(2(-1)) \\\\\\ 9x^2-12x+4+2\implies 9x^2-12x+6 \\\\[-0.35em] ~\dotfill\\\\ a(9x^2-12x+6)=y\hspace{5em}\stackrel{\textit{now let's make}}{a=-\frac{1}{9}} \\\\\\ -\cfrac{1}{9}(9x^2-12x+6)=y\implies \boxed{-x^2+\cfrac{4}{3}x-\cfrac{2}{3}=y}[/tex]

Answer:

[tex]\dfrac{4096\pi}{5}\approx 2573.593\; \sf (3\;d.p.)[/tex]

Step-by-step explanation:

The shell method is a calculus technique used to find the volume of a solid revolution by decomposing the solid into cylindrical shells. The volume of each cylindrical shell is the product of the surface area of the cylinder and the thickness of the cylindrical wall. The total volume of the solid is found by integrating the volumes of all the shells over a certain interval.

The volume of the solid formed by revolving a region, R, around a vertical axis, bounded by x = a and x = b, is given by:

[tex]\displaystyle 2\pi \int^b_ar(x)h(x)\;\text{d}x[/tex]

where:

[tex]\hrulefill[/tex]

We want to find the volume of the solid formed by rotating the region bounded by y = 0, y = √x, x = 0 and x = 16 about the y-axis.

As the axis of rotation is the y-axis, r(x) = x.

Therefore, in this case:

[tex]r(x)=x[/tex]

[tex]h(x)=\sqrt{x}[/tex]

[tex]a=0[/tex]

[tex]b=16[/tex]

Set up the integral:

[tex]\displaystyle 2\pi \int^{16}_0x\sqrt{x}\;\text{d}x[/tex]

Rewrite the square root of x as x to the power of 1/2:

[tex]\displaystyle 2\pi \int^{16}_0x \cdot x^{\frac{1}{2}}\;\text{d}x[/tex]

[tex]\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x[/tex]

Integrate using the power rule (increase the power by 1, then divide by the new power):

[tex]\begin{aligned}\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x&=2\pi \left[\dfrac{2}{5}x^{\frac{5}{2}}\right]^{16}_0\\\\&=2\pi \left[\dfrac{2}{5}(16)^{\frac{5}{2}}-\dfrac{2}{5}(0)^{\frac{5}{2}}\right]\\\\&=2 \pi \cdot \dfrac{2}{5}(16)^{\frac{5}{2}}\\\\&=\dfrac{4\pi}{5}\cdot 1024\\\\&=\dfrac{4096\pi}{5}\\\\&\approx 2573.593\; \sf (3\;d.p.)\end{aligned}[/tex]

Therefore, the volume of the solid is exactly 4096π/5 or approximately 2573.593 (3 d.p.).

[tex]\hrulefill[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Power Rule of Integration}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}(+\;\text{C})$\\\end{minipage}}[/tex]

The value of the triple integral[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex] by using spherical coordinates [tex]2\pi(e^{-1}-e^{-9})[/tex].

Given that the triple integral is-

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

E is the region bounded by the spheres which are,

[tex]x^2+y^2+z^2=1\\\\x^2+y^2+z^2=9[/tex]

In spherical coordinates we have,

x = r cosθ sin ∅

y = r sinθ sin∅

z = r cos∅

dV = r²sin∅ dr dθ d∅

E contains two spheres of radius 1 and 3 () respectively, the bounds will be like this,

1 ≤ r ≤ 3

0 ≤ θ ≤ 2π

0 ≤ ∅ ≤ π

Then

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

[tex]\int\int\int _{E} \frac{e^{-r^2}}{r}r^2Sin\phi drd\phi d\theta\\\\2\pi \int_{0}^{\pi} \int_1^3 re^{-r^2} dr d\phi\\\\2\pi \int_1^3 re^{-r^2} dr\\\\2\pi(e^{-1}-e^{-9})[/tex]

The complete question is-

Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.

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Answer:

We have two equations:

√x +2y^2 = 15 ----(1)

√4x - 4y^2=6 ----(2)

Let's solve for x:

From (1), we have:

√x = 15 - 2y^2

Squaring both sides, we get:

x = (15 - 2y^2)^2

Expanding, we get:

x = 225 - 60y^2 + 4y^4

From (2), we have:

√4x = 6 + 4y^2

Squaring both sides, we get:

4x = (6 + 4y^2)^2

Expanding, we get:

4x = 36 + 48y^2 + 16y^4

Substituting the expression for x from equation (1), we get:

4(225 - 60y^2 + 4y^4) = 36 + 48y^2 + 16y^4

Simplifying, we get:

900 - 240y^2 + 16y^4 = 9 + 12y^2 + 4y^4

Rearranging, we get:

12y^2 - 12y^4 = 891

Dividing both sides by 12y^2, we get:

1 - y^2 = 74.25/(y^2)

Multiplying both sides by y^2, we get:

y^2 - y^4 = 74.25

Let z = y^2. Substituting, we get:

z - z^2 = 74.25

Rearranging, we get:

z^2 - z + 74.25 = 0

Using the quadratic formula, we get:

z = (1 ± √(1 - 4(1)(74.25))) / 2

z = (1 ± √(-295)) / 2

Since the square root of a negative number is not real, there are no real solutions for z, which means there are no real solutions for y and x.

Therefore, the answer is "no solution".

Sin 30 =3/x

1/2=3/x

x=6