Solve |y + 2| > 6

{y|y < -8 or y > 4}
{y|y < -6 or y > 6}
{y|y < -4 or y > 4}

The scale factor of this dilation is 2/3.

It is required to find  scale factor of this dilation.

Scale Factor is defined as the ratio of the size of the new image to the size of the old image.

In the figure showing 6 to 9 is 2/3 dilation and 10 to 15 is also a 2/3 dilation.

So, the scale factor of this dilation is 2/3.

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Option b is correct. The scale factor is [tex]1 \frac{1}{2}[/tex].

To determine the scale factor of the dilation from Triangle ABC to Triangle A'B'C', we need to compare the lengths of corresponding sides.

The side lengths of Triangle ABC are:

The side lengths of Triangle A'B'C' are:

The scale factor is calculated by dividing the lengths of the corresponding sides of the triangles. Let's use AB and A'B' for our calculation:

Scale Factor = A'B'/AB = 9/6 = 3/2 = 1 whole 1/2

Thus, the scale factor is [tex]1 \frac{1}{2}[/tex], which corresponds to option b.

Complete question:

What is the scale factor of this dilation? Triangle ABC to A'B'C'.

Triangle ABC with AB = 6, BC = 10, CA = 6

Triangle A'B'C' with A'B'= 9, B'C'= 15, C'A' =9

a. 2/3

b. [tex]1 \frac{1}{2}[/tex]

c. 3

d. 5

Answer: 4 units.

Step-by-step explanation:

Once each point is plotted, you get the rectangle attached.

You can observe in the figure that the base of the rectangle is its longer side.

 Since the length of the base of the rectangle goes from [tex]x=-1[/tex] to [tex]x=3[/tex], you can say that the lenght of the base is the difference between 3 and -1.

So, you need to subtract this two coordinates, getting that the lenght of the base of the rectangle attached is:

[tex]lenght_{(base)}=3-(-1)\\\\lenght_{(base)}=3+1\\\\lenght_{(base)}=4\ units[/tex]

Answer: 4 units

Step-by-step explanation:

Answer:

[tex]\text{1) }0.1141814[/tex]

[tex]\text{2) }0.6193473[/tex]

[tex]\text{3) }0.2003702[/tex]

Step-by-step explanation:

[tex]\text{1) }P(-0.3203\leq Z \leq-0.0287)=P(Z\leq -0.0287)-P(Z\leq-0.3203)\\\\=0.4885519-0.3743705\\\\=0.1141814[/tex]

[tex]\text{2) }P( -0.5156 \leq Z \leq1.4215)=P(Z\leq 1.4215)-P(Z\leq -0.5156 )\\\\=0.9224142-0.3030669\\\\=0.6193473[/tex]

[tex]\text{3) }P(0.1269\leq Z \leq0.6772)=P(Z\leq 0.6772)-P(Z\leq-0.1269)\\\\=0.7508604- 0.5504902\\\\=0.2003702[/tex]

Answer:

Zero

Step-by-step explanation:

We are given the following expression and we are to determine what is the sign of its product:

[tex] 4 . 3 . ( - 3 . 2 ) . 0 [/tex]

One of the three terms in the expression is positive while one is negative. So if we start multiplying the two terms from the left side. we will get a negative number.

But when we will multiply it with zero, the whole product will become zero as anything times zero is always zero. Therefore, answer will be zero.

Answer:

it is negative

Step-by-step explanation:

a positive times a negative is a negative.

For this case we have the following expression:

[tex]y^ 7 * y^ 9 =[/tex]

By definition of multiplication of powers of the same base, we have to put the same base and add the exponents, that is:

[tex]a ^ n * a ^ m = a ^ {n + m}[/tex]

So:

[tex]y ^ 7 * y ^ 9 = y ^{7 + 9} = y ^ {16}[/tex]

Answer:

[tex]y^{16}[/tex]

Sin is the measure of the opposite leg over the hypotenuse from the given angle:

opposite/hypotenuse

We must find the sin of Angle A, and in order to do so we must find the opposite leg and hypotenuse:

opposite leg/hypotenuse

8/10

Simplify:

8/10 = 4/5

Hence, the sin of <A is 4/5

For this case we have by definition, the sine of an angle is given by the leg opposite the angle on the hypotenuse of the triangle. Then, according to the figure we have:

[tex]Sin (A) = \frac {8} {10}[/tex]

Simplifying we have to:

[tex]Sin (A) = \frac {4} {5}[/tex]

Answer:

Option B

Answer:

The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is __-80___feet per second.

Step-by-step explanation:

The average change rate m is calculated using the following formula

[tex]m=\frac{f(t_2)-f(t_1)}{t_2-t_1}[/tex]

In this case [tex]f(t) = -16t^2 + 48t + 100[/tex],  [tex]t_2 = 5\ s\ \ , t_1=3\ s[/tex]

Then

[tex]f(t_2) = f(5) =-16(5)^2 + 48(5) + 100[/tex]

[tex]f(t_2) = -60[/tex]

[tex]f(t_1) = f(3) =-16(3)^2 + 48(3) + 100[/tex]

[tex]f(t_1) = 100[/tex]

Finally

[tex]m=\frac{(-60)-100}{5-3}[/tex]

[tex]m=-80[/tex]

Answer:

{P|P < -3 or P > 3}

Step-by-step explanation:

When we remove the absolute value bars, we take the positive value, and then flip the inequality and take the negative.  Since the original equation is a greater than, we use the or

p > 3   or    p < -3

Answer:

[tex]\large\boxed{\{P\ |\ P<-3\ or\ P>3\}}[/tex]

Step-by-step explanation:

[tex]\text{The absolute value:}\\\\|a|=\left\{\begin{array}{ccc}a&for&a\geq0\\-a&for&a<0\end{array}\right[/tex]

[tex]|P|>3\iff P>3\ or\ P<-3[/tex]

Answer:

ab + 3a + 8b + 24

Step-by-step explanation:

(a + 8)(b + 3)

a(b + 3) + 8(b + 3)

ab + 3a + 8b + 24

Answer:

[tex]\large\boxed{15\div6\dfrac{2}{3}=2\dfrac{1}{4}}[/tex]

Step-by-step explanation:

[tex]15\div6\dfrac{2}{3}\\\\\text{convert the mixed number to the improper fraction:}\ 6\dfrac{2}{3}=\dfrac{6\cdot3+2}{3}=\dfrac{20}{3}\\\\15\div\dfrac{20}{3}=15\cdot\dfrac{3}{20}=15\!\!\!\!\!\diagup^3\cdot\dfrac{3}{20\!\!\!\!\!\diagup_4}=\dfrac{(3)(3)}{4}=\dfrac{9}{4}=2\dfrac{1}{4}[/tex]

For this case we must rewrite the mixed number as a fraction:

[tex]6 \frac {2} {3} = \frac {3 * 6 + 2} {3} = \frac {20} {3}[/tex]

Rewriting the expression we have:

[tex]\frac {\frac {15} {1}} {\frac {20} {3}} =[/tex]

Applying double C we have:

[tex]\frac {3 * 15} {20 * 1} =\\\frac {45} {20} =[/tex]

We simplify dividing by 5 the numerator and denominator:

[tex]\frac {9} {4}[/tex]

Converting to mixed number we have:

[tex]2 \frac {1} {4}[/tex]

Answer:

[tex]\frac {9} {4} = 2 \frac {1} {4}[/tex]

Answer: 1.8446

Step-by-step explanation:

Given claim : [tex]\mu>\mu_0,\text{ where }\mu_0=135[/tex]

Sample size : [tex]n=23[/tex]

Sample mean : [tex]\overline{x}=140[/tex]

Standard deviation : [tex]\sigma = 13[/tex]

The test statistic for population mean is given by :-

[tex]z=\dfrac{x-\mu_0}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

[tex]\Rightarrow\ z=\dfrac{140-135}{\dfrac{13}{\sqrt{23}}}\\\\\Rightarrow\ z=1.84455058589\approx1.8446[/tex]

Hence, the value of test statistic =  1.8446

Answer:

378.5 or just 378

Step-by-step explanation:

This is a linear model with x representing the number of generations that's gone by, y is the number of butterflies after x number of generations has gone by, and the 350 represents the number of butterflies initially (before any time has gone by.  When x = 0, y = 350 so that's the y-intercept of our equation.)

The form for a linear equation is y = mx + b, where m is the rate of change and b is the y-intercept, the initial amount when x = 0.

Our rate of change is 1.5 and the initial amount of butterflies is 350, so filling in the equation we get a model of y = 1.5x + 350.

If we want y when x = 19, plug 19 in for x and solve for y:

y = 1.5(19) + 350

y = 378.5

Since we can't have .5 of a butterfly we will round down to 378

Answer:

[tex]y=x^2+2x-8[/tex]

Step-by-step explanation:

When you graph those points on a piece of graph paper it appears that the points are in the form of a positive x^2 parabola, which has the standard form

[tex]y=ax^2+bx+c[/tex]

We just need to solve for a, b, and c.  Easy.  We have 3 points from the table.  We will use all three of them to find the values of a, b, and c.

Use the points (0, -8), (2, 0), and (4, 16).  You can use any points, but I chose the one with an x value of 0 for a good reason, and chose the other 2 because I don't like too many negatives!

Use the first point in those above to solve for c:

[tex]-8=a(0)^2+b(0)+c[/tex]

From this you solve for c:  c = -8

Now use the next point along with the value of c to find another equation:

[tex]0=a(2)^2+b(2)-8[/tex] and

[tex]0=4a+2b-8[/tex] so

8 = 4a + 2b

That equation will be used again in a minute.

Use the last point to solve for yet another equation (stay with me...we are almost there!):

[tex]16=a(4)^2+b(4)-8[/tex] and

24 = 16a + 4b

Now we will use the method of elimination to solve for b:

8  =  4a  +  2b

24 = 16a + 4b

Multiply the first equation by -4 to eliminate the a terms:

-32 = -16a - 8b

24 = 16a + 4b

leaves you with

-4b = -8 and b = 2.  Now plug that back in to solve for a:

If 8 = 4a + 2b, then 8 = 4a + 2(2) and 8 = 4a + 4

4a = 4 and a = 1

Again, your equation is

[tex]y=x^2+2x-8[/tex]

Answer:  The mean and variance of the wavelength distribution for this radiation are 642.5 nm and 75 nm.

Step-by-step explanation:

The mean and variance of a continuous uniform distribution function with parameters m and n is given by :-

[tex]\text{Mean=}\dfrac{m+n}{2}\\\\\text{Variance}=\dfrac{(n-m)^2}{12}[/tex]

Given : [tex]m=625\ \ \ n=655[/tex]

[tex]\text{Then, Mean=}\dfrac{625+655}{2}=642.5\\\\\text{Variance}=\dfrac{(655-625)^2}{12}=75[/tex]

Hence, the mean and variance of the wavelength distribution for this radiation are 642.5 nm and 75 nm.

Answer: 0.0051

Step-by-step explanation:

Given: Mean : [tex]\mu = 50\text{ inch}[/tex]

Standard deviation : [tex]\sigma =1.1\text{ inch}[/tex]

Sample size : [tex]n=8[/tex]

The formula to calculate z is given by :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x= 51

[tex]z=\dfrac{51-50}{\dfrac{1.1}{\sqrt{8}}}=2.57129738613\approx2.57[/tex]

The P Value =[tex]P(Z>51)=P(z>2.57)=1-P(z<2.57)=1-0.994915=0.005085\approx0.0051[/tex]

Hence, the probability that the sample mean hardness for a random sample of 8 pins is at least 51 =0.0051

Answer:

  f(x) = x^4 +x^3 -2x^2 -8x +9

Step-by-step explanation:

You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...

  f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant

The second antiderivative is ...

  f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant

Then the constants can be found from ...

  f(0) = q = 9

  f(1) = 1 + 1 - 2 +p + 9 = 1

  p = -8

The solution is ...

  f(x) = x^4 +x^3 -2x^2 -8x +9

_____

The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.

Final answer:

To find f(x) given f''(x) = 12x² + 6x − 4, one must integrate twice and use the initial conditions f(0) = 9 and f(1) = 1 to solve for the constants. The final function is f(x) = x⁴ + x³ - 2x² - 8x + 9.

Explanation:

The question asks to find the antiderivative f(x) given its second derivative f''(x) =  12x² + 6x − 4, and two initial conditions, f(0) = 9, and f(1) = 1. To solve for f(x), we first integrate the second derivative twice to get the original function.

Integrating f''(x), we get:

f'(x) = ∫( 12x² + 6x - 4)dx = 4x³ + 3x² - 4x + C

We then integrate f'(x) to find f(x):

f(x) = ∫(4x³ + 3x² - 4x + C)dx = x⁴ + x³ - 2x² + Cx + D

Using the initial conditions:

Therefore, the original function is f(x) =  x⁴ + x³ - 2x² - 8x + 9.

Answer:

x is less than three-fourths

x is less than 0.75

Step-by-step explanation:

First distribute the 3 on the right side of the equation.

-2x + 3 > 3(2x-1)

-2x + 3 > 6x -3

Then you transpose and combine like terms

-2x -6x > -3 - 3

-8x > -6

Divide both sides by -8, but if you will do this, you need to remember if you divide both sides by a negative number, you need to swap the inequality.

x < 3/4

x < 0.75

Answer:

x is less than three-fourths

x is less than 0.75

Step-by-step explanation:

Two boys working together can paint a fence in 5 hours with a work rate of 0.2 fences per hour. Adding one more boy increases this work rate to 0.3 fences per hour. This would allow them to complete the painting of the fence in approximately 3.3 hours.

This problem can be solved using the concept of work rate. The work rate is defined as the amount of work done per unit time.

In this case, two boys can paint a fence in 5 hours. So, their combined work rate is 1 fence per 5 hours, or 0.2 fences per hour.

When we add another boy to the group, we increase the total work rate by 50% as now there are 3 boys. So, their combined work rate becomes 0.2 fences/hour + (0.2 fences/hour) * 50% = 0.3 fences/hour.

To find out how long it would take these three boys to paint the fence, we divide the total work (1 fence) by the total work rate (0.3). So, 1 fence divided by 0.3 fences/hour = approximately 3.3 hours. That's how long it would take three boys to paint the fence.

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The likelihood of a randomly chosen woman having a red blood cell count higher than the typical range of 4.2 to 5.4 million cells per microliter, given that the counts are normally distributed with a mean of 4.577 and a standard deviation of 0.382 million cells, is approximately 0.0158 or 1.58% when expressed as a percentage.

The subject matter here is the use of statistics to understand biological phenomena, specifically the distribution of red blood cell counts in women. The question asks for the probability that a randomly selected woman has a red blood cell count above the normal range of 4.2 to 5.4 million cells per microliter, given that the counts are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells.

Firstly, to answer this question, we must establish the z-scores for the boundaries of our range. The z-score formula is Z = (X - μ) / σ, where X is the value we are evaluating, μ is the mean, and σ is the standard deviation. The upper boundary of our range is 5.4 million cells, so to find the z-score for this we substitute into the formula: Z = (5.4 - 4.577) / 0.382, which gives us a Z-score of approximately 2.15.

However, we are interested in the probability of a woman having a count above the normal range, so we need the area of the curve beyond this z-score. You can find this probability using standard normal distribution tables or a calculator, which suggests that the probability of having a count above 5.4 is approximately 0.0158, or 1.58% when expressed as a percentage and rounded to four decimal places.

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Answer:

63.16 in approx.

Step-by-step explanation:

Let the shorter leg be S.  Then the longer leg is L = 3S + 3.

The formula for the area of a triangle is A = (1/2)(base)(height).  Here, that works out to A = 84 in^2 = (1/2)(S)(3S + 3).

Simplifying, we get 168 in^2 = S(3S + 3), or

3S^2 + 3S - 168 = 0, or

 S^2  +  S  - 56   = 0.  This factors as follows:  (S - 8)(S + 7) = 0, so the positive root is S = 8.  We discard the negative root.

Thus, the shorter leg length is 8 and the longer leg length is 3(8) + 3, or 27.

According to the Pythagorean Theorem, the hypotenuse length is given by

L^2 = 8^2 + 27^2, or

L^2 = 64 + 729 = 793.

L = hypotenuse length = √793, or approx. 28.2 in.

Then the perimeter of the triangle is 8 + 27 + 28.2 in, or approx. 63.16 in

The cost of toolset before the discount was applied is:

                    Rs.  521.5189

It is given that:

A tool set has been discounted down to a price of Rs 412.00.

The percent discount that is provided to us is: 21%

Let the actual price of toolset be: Rs. x

This means that:

[tex]x-21\%\ of x=412\\\\i.e.\\\\x-0.21x=412\\\\i.e.\\\\(1-0.21)x=412\\\\i.e.\\\\0.79x=412\\\\i.e.\\\\x=\dfrac{412}{0.79}\\\\x=521.5189[/tex]

   Hence, the actual price of toolset i.e. cost before discount is:

                        Rs.   521.5189

To find the original price of the toolset before the discount was applied, we can set up an equation and solve for the original price. The original price of the toolset was R520.89.

To find the original price of the toolbox, we need to first calculate the amount of the discount. The discount given is 21%. Let's say the original price of the toolbox is P. We can calculate the amount of the discount by multiplying the original price by the discount percentage:

Discount = P * 21% = P * 0.21

The discounted price is given as R412.00. So we can set up the equation:

Original price - Discount = Discounted price

P - P * 0.21 = R412.00

We can now solve for P:

P(1 - 0.21) = R412.00

0.79P = R412.00

P = R412.00 / 0.79

P = R520.89

Therefore, the original price of the toolset before the discount was applied was R520.89.

Answer:

the answer is a

Step-by-step explanation:

i just know

Answer: A.

Step-by-step explanation:

Equation: V=lwh

5 × 10 × 15 = 750 cubic cm

Answer: 640

Step-by-step explanation:

Since the two triangles are similar we can simply multiply the lesser triangle's area by a constant to get our answer.

Polygon FGHIJ is ABCDE with a scale change of 4

For the reason that we are dealing with area, we will multiply 40 by 4² in stead of just 4.

40 * 16 = 640

Answer:

B. 640

Step-by-step explanation:

got it right 2021

Answer:

Two lines are parallel when they share the same slope.

Step-by-step explanation:

Two lines are parallel when they share the same slope.

The slope-intercept form of the equation of a line is: y=mx + b, where 'm' is the slope and 'b' the y-intercept.

If two equations have the same value for 'm', then those lines are parallel, for example:

y = 3x + 8 (Red line)

y = 3x + 5 (Blue line)

y = 3x - 10 (Green line)

All the equations stated above are parallel, to show that, I'm attaching the graph of the equations :).

Final answer:

The most B. reasonable interpretation of a 90% chance of rain is that there is a 0.90 probability that it will rain somewhere in the region.

Explanation:

The most reasonable interpretation of a 90% chance of rain, according to the given weather forecasting website, is option B: There is a 0.90 probability that it will rain somewhere in the region at some point during the day. This means that there is a high likelihood that rain will occur in the region, but it does not guarantee that every part of the region will experience rain. It indicates that out of 100 instances, rain is expected in approximately 90 of them.

It is important to note that options A, C, and D are not reasonable interpretations because option A assumes that 100% of the region will get rain, option C assumes that it will rain for 90% of the day, and option D states that none of the interpretations are reasonable, which is not accurate.

Final answer:

The most reasonable interpretation of a 90% chance of rain in a weather forecast is that there is a 0.90 probability of rainfall somewhere in the specified region at some point during the day.

Explanation:

When a weather forecast indicates a 90% chance of rain, it means there is a 0.90 probability that it will rain somewhere in the specified region at some point during the day. Therefore, the correct interpretation based on the given options is B. There is a 0.90 probability that it will rain somewhere in the region at some point during the day. Interpretation A, suggesting that 90% of the region will get rain, is not accurate because the percentage given in a forecast refers to probability, not an area's coverage. Interpretation C, suggesting it will rain for 90% of the day, is also incorrect because the percentage does not refer to the duration of rain but to the probability of occurrence. Statement D is incorrect because B provides a reasonable interpretation.

I guess the "5" is supposed to represent the integral sign?

[tex]I=\displaystyle\int_1^4\ln t\,\mathrm dt[/tex]

With [tex]n=10[/tex] subintervals, we split up the domain of integration as

[1, 13/10], [13/10, 8/5], [8/5, 19/10], ... , [37/10, 4]

For each rule, it will help to have a sequence that determines the end points of each subinterval. This is easily, since they form arithmetic sequences. Left endpoints are generated according to

[tex]\ell_i=1+\dfrac{3(i-1)}{10}[/tex]

and right endpoints are given by

[tex]r_i=1+\dfrac{3i}{10}[/tex]

where [tex]1\le i\le10[/tex].

a. For the trapezoidal rule, we approximate the area under the curve over each subinterval with the area of a trapezoid with "height" equal to the length of each subinterval, [tex]\dfrac{4-1}{10}=\dfrac3{10}[/tex], and "bases" equal to the values of [tex]\ln t[/tex] at both endpoints of each subinterval. The area of the trapezoid over the [tex]i[/tex]-th subinterval is

[tex]\dfrac{\ln\ell_i+\ln r_i}2\dfrac3{10}=\dfrac3{20}\ln(ell_ir_i)[/tex]

Then the integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac3{20}\ln(\ell_ir_i)\approx\boxed{2.540}[/tex]

b. For the midpoint rule, we take the rectangle over each subinterval with base length equal to the length of each subinterval and height equal to the value of [tex]\ln t[/tex] at the average of the subinterval's endpoints, [tex]\dfrac{\ell_i+r_i}2[/tex]. The area of the rectangle over the [tex]i[/tex]-th subinterval is then

[tex]\ln\left(\dfrac{\ell_i+r_i}2\right)\dfrac3{10}[/tex]

so the integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac3{10}\ln\left(\dfrac{\ell_i+r_i}2\right)\approx\boxed{2.548}[/tex]

c. For Simpson's rule, we find a quadratic interpolation of [tex]\ln t[/tex] over each subinterval given by

[tex]P(t_i)=\ln\ell_i\dfrac{(t-m_i)(t-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+\ln m_i\dfrac{(t-\ell_i)(t-r_i)}{(m_i-\ell_i)(m_i-r_i)}+\ln r_i\dfrac{(t-\ell_i)(t-m_i)}{(r_i-\ell_i)(r_i-m_i)}[/tex]

where [tex]m_i[/tex] is the midpoint of the [tex]i[/tex]-th subinterval,

[tex]m_i=\dfrac{\ell_i+r_i}2[/tex]

Then the integral [tex]I[/tex] is equal to the sum of the integrals of each interpolation over the corresponding [tex]i[/tex]-th subinterval.

[tex]I\approx\displaystyle\sum_{i=1}^{10}\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt[/tex]

It's easy to show that

[tex]\displaystyle\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt=\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)[/tex]

so that the value of the overall integral is approximately

[tex]I\approx\displaystyle\sum_{i=1}^{10}\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)\approx\boxed{2.545}[/tex]

The question asks to approximate the given integral using three numerical methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. These methods use simple geometric shapes to estimate the area under the curve. Due to the complexity of the integral in question, assistance from computer software or a graphing calculator will likely be necessary.

The question is about using numerical methods to approximate a given integral using three methods: the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule. All of these methods are used to approximate the definite integral of a function over an interval. They divide the interval into n subintervals and then use simple geometric shapes to approximate the area under the curve of the function.

To compute these, you would follow these steps: 1. For the Trapezoidal Rule, average the end points and multiply by the width of each interval. 2. For the Midpoint Rule, evaluate the function at the midpoint of each interval, multiply by the width of each interval. 3. For Simpson's Rule, apply the specific weighted average formula that gives more weight to the midpoint

Please note, however, that due to the complexity of the integral of ln(t), you would likely need to use computer software or a graphing calculator to perform these approximations. Please consult with your teacher for the best approach based on what resources are available to you.

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Hello There!

[tex]\frac{5}{100}[/tex] written as a decimal is 0.05

Step #1     5/100 is the same thing as 5/5 over 100/5

Step #2 you have a quotient of 1/20

Step #3 divide 1 by 20 and you get a quotient of 0.05

In the given problem, 0.05 is the fraction [tex]\frac{5}{100}[/tex] written as a decimal.

A fraction is a mathematical expression that represents a part or a division of a whole. It is used to represent numbers that are not whole numbers or integers. A fraction consists of two components:

1. Numerator: The numerator is the number on the top of the fraction. It represents the quantity or part of the whole being considered.

2. Denominator: The denominator is the number at the bottom of the fraction. It represents the total number of equal parts into which the whole is divided.

To convert the fraction [tex]\frac{5}{100}[/tex] to a decimal, you can simply divide the numerator, 5 by the denominator, 100.

5 [tex]\div[/tex] 100 = 0.05.

Therefore, [tex]\frac{5}{100}[/tex] is equal to 0.05 as a decimal.

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Answer:

Max = 86; min = 36.54

Step-by-step explanation:

[tex]f(x) = x^{2} + \dfrac{85}{x}[/tex]

Step 1. Find the critical points.

(a) Take the derivative of the function.

[tex]f'(x) = 2x - \dfrac{85}{x^{2}}[/tex]

Set it to zero and solve.

[tex]\begin{array}{rcl}2x - \dfrac{85}{x^{2}} & = & 0\\\\2x^{3} - 85 & = & 0\\2x^{3} & = & 85\\\\x^{3} & = &\dfrac{85}{2}\\\\x & = & \sqrt [3]{\dfrac{85}{2}}\\\\& \approx & 3.490\\\end{array}\[/tex]

(b) Calculate ƒ(x) at the critical point.  

[tex]f(3.490) = 3.490^{2} + \dfrac{85}{3.490} = 12.18 + 24.36 = 36.54[/tex]

Step 2. Calculate ƒ(x) at the endpoints of the interval

[tex]f(1) = 1^{2} + \dfrac{85}{1} = 1 + 85 = 86\\\\f(5) = 5^{2} + \dfrac{85}{5} = 25 + 17 = 42[/tex]

Step 3.Identify the maxima and minima.

ƒ(x) achieves its absolute maximum of 86 at x = 1 and its absolute minimum of 36.54 at x = 3.490

The figure below shows the graph of ƒ(x) from x = 1 to x = 5.

The maximum and minimum values of the mathematical function f(x) = x^2 + 85/x on the interval [1, 5] occur at the endpoints with the maximum being 86 at x=1 and minimum being 37 at x=5.

To find the maximum and minimum values of the function f(x) = x2 + 85/x over the interval [1, 5], we first find the critical points in that interval. The critical points occur where the derivative of the function is zero or undefined. The derivative of the function f(x) is 2x - 85/x2 and it is undefined at x = 0 and becomes 0 at x = sqrt (42.5).

Since x = 0 is not in our interval, we disregard it. The value x = sqrt (42.5) is also outside our interval [1, 5], so we disregard this too.

Therefore, the maximum and minimum values of the function on the interval [1, 5] occur at the endpoints. We substitute these endpoint values into the function:

Therefore, the maximum value is 86 and the minimum value is 37, at x equals 1 and 5, respectively.

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