Describe where the function has a vertical asymptote and EXPLAIN how you found your answer.

[tex]f(x) \frac{x^2 +7x+10}{x^2+9x+20}[/tex]

Answer:

The radius of the oblique cylinder is 4 cm.

Step-by-step explanation:

Given:

The volume of the oblique cylinder = [tex]192 \pi cm^3[/tex]

Height of the cylinder = 12 cm

To Find:

Radius of the oblique cylinder = ?

Solution:

we know that the volume of the oblique cylinder  

=>[tex]\text{base area of the cylinder} \times \text{height of the cylinder}[/tex]------------------------------(1)

where base area of the cylinder is the area of the circle

so area of the circle = base area

[tex]\pi r^2[/tex] = base area---------------(2)

Substituting (2) in (1)

[tex]192 \pi = \pi r^2 \times height[/tex]

[tex]192 \pi = \pi \times r^2 \times height[/tex]

[tex]192 \pi = \pi \times r^2 \times 12[/tex]

[tex]\frac{192 \pi}{ \pi \times 12}= r^2[/tex]

[tex]\frac{192}{12}= r^2[/tex]

[tex]16= r^2[/tex]

[tex] r^2 =16[/tex]

[tex] r =\sqrt{16}[/tex]

r=4

Answer:  The required length of the diameter of the circumscribed circle is 24 units.

Step-by-step explanation:  Given that one side of a triangle has length 12 units and its opposite angle measures 30 degrees.

We are to find the diameter of the circumscribed circle.

We know that

if a, b, c are the lengths of the three sides of a triangle and A, B, C are the corresponding measures of the opposite angles respectively, then the ratio

[tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=d,[/tex]

is said to the length of the diameter of the circumscribed circle of the triangle.

According to the given information, we have

a = 12  and  A = 30°.

Therefore, the length of the diameter of the circumscribed circle is

[tex]d=\dfrac{a}{\sin A}=\dfrac{12}{\sin 30^\circ}=\dfrac{12}{\frac{1}{2}}=24.[/tex]

Thus, the required length of the diameter of the circumscribed circle is 24 units.

Given the details of a triangle with  one of its sides as 12 and opposite angle as 30 degrees, the diameter of the circumscribed circle is twice the length of thegiven side. Since the hypotenuse of the triangle is the diameter of a circumscribed circle, the diameter in this case is 24.

The subject of this question is pertaining to geometry, specifically, the relationship between sides and angles in triangles, and circles. In particular, we are dealing with a situation where we have a triangle inscribed in a circle (a triangle with a circumscribed circle), and we are asked to find the diameter of the said circle.

Now, from trigonometry, we know that the sine of any angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. In this case, you're given that one side of the triangle (which we'll assume is the opposite side) is 12, and its opposite angle is 30 degrees. Since the sine of 30 degrees is 0.5, this means that the hypotenuse of this triangle is actually twice the length of the given side. Thus, the hypotenuse is 2 * 12 = 24.

This is significant because, in any triangle inscribed in a circle, the diameter of the circle is equal to the length of the hypotenuse. Therefore, in this case, the diameter of the circle would be 24.

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The sum of the given series [tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex]  is 130.

Step-by-step explanation:

The sum of series need to be found is [tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] .

Apply sum rule,

[tex]\sum x_n+y_n=\sum x_n+\sum y_n[/tex]

[tex]\sum _{a=1}^{10}\:\left(2a+2\right).[/tex] =[tex]\sum 2a+\sum 2[/tex].

Apply the constant multiplication rule,

[tex]\sum c\cdot a_n=c\cdot \sum a_n[/tex] .

[tex]\sum 2\cdot a=2\cdot \sum a[/tex].

[tex]\sum _{a=1}^{10}n[/tex] = 1+2+3+4+5+6+7+8+9+10.

[tex]\sum _{a=1}^{10}n[/tex] = 55.

[tex]\sum 2\cdot a[/tex]=2×55.

[tex]\sum 2\cdot a [/tex]=110.

To find [tex]\sum _{a=1}^{10}2[/tex],

Apply sum rule, [tex]\sum _{k=1}^n\:a\:=\:a\cdot n[/tex] ,

[tex]\sum _{a=1}^{10}2[/tex] =2×10.

[tex]\sum _{a=1}^{10}2[/tex] = 20.

[tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] = 110+20.

[tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] =130.

Answer:

7 software packages

Step-by-step explanation:

Given,

Signing amount = $ 225,

Additional amount for each software package = $ 25

Thus, the total amount for x software packages = signing amount + additional amount for x packages

= 225 + 25x

If total amount ≥ $ 400

225 + 25x ≥ 400

25x ≥ 400 - 225

25x ≥ 175

[tex]\implies x \geq \frac{175}{25}=7[/tex]

Hence, the minimum number of software packages that she must sell would be 7.

Final answer:

Alison needs to sell at least 7 software packages this week on top of her base salary to earn at least $400.

Explanation:

Alison wants to earn at least $400 this week by selling software packages. She earns a base $225 per week plus $25 for each software package sold. To determine the minimum number of software packages Alison must sell, we subtract her base salary from her desired earnings for the week:

Required earnings (>=$400) - Base salary ($225) = Sales required from software packages

$400 - $225 = $175

Now divide the sales required from software packages by the amount earned per package:

$175 / $25 = 7

Therefore, Alison needs to sell at least 7 software packages to meet her goal of $400

The shape and bond angle of oxy-nitrogen ions is determined by the number of bonds and lone pairs around the central atom, with examples being trigonal pyramidal for nitrogen with three bonds and one lone pair, tetrahedral for carbon with four bonds, and bent for oxygen with two bonds and two lone pairs.

The oxynitrogen ions have specific shapes and bond angles that correlate with their electron-pair geometries. For example, a nitrogen atom with three bonds and one lone pair will have a trigonal pyramidal shape with bond angles close to 109°. Carbon atoms in different environments also exhibit varied shapes: a carbon atom in CH₂ with four bonds and no lone pairs has a tetrahedral shape with bond angles of 109.5°, while a carbon atom in CO₂ with two double bonds (treated as equivalent to three bonds) and no lone pairs forms a trigonal planar shape.

An oxygen atom in OH with two bonds and two lone pairs has a bent or angular shape with bond angles close to 109°. It's essential to note that electron pair repulsion and hybridization of orbitals play a crucial role in determining these geometric shapes and the bond angles.

Based on the options provided and the common shapes of oxynitrogen ions, the correct answer would be:

- Trigonal planar, 120°

- Tetrahedral, 109.5°

- Bent, 120°

- Pyramidal, 109°

These shapes and bond angles are commonly observed in oxynitrogen ions depending on the arrangement of atoms and lone pairs around the central nitrogen atom.

The shape and bond angle of oxynitrogen ions depend on the arrangement of atoms and lone pairs around the central nitrogen atom. Let's go through each option:

1. Linear, 180°: This shape occurs when there are no lone pairs on the central nitrogen atom, and there are two bonded atoms around it. The bond angle is indeed 180°.

2. Trigonal planar, 120°: This shape occurs when there is one lone pair and three bonded atoms around the central nitrogen atom. The bond angle is indeed 120°.

3. Tetrahedral, 109.5°: This shape occurs when there are four bonded atoms around the central nitrogen atom and no lone pairs. The bond angle is indeed 109.5°.

4. Bent, 120°: This shape occurs when there is one lone pair and two bonded atoms around the central nitrogen atom. The bond angle is close to 120° but may deviate slightly due to lone pair-bond pair repulsions.

5. Bent, 109°: This shape is less common for oxynitrogen ions but can occur if there is a lone pair and two bonded atoms around the central nitrogen atom. The bond angle would be closer to 109° due to lone pair-bond pair repulsions.

6. Pyramidal, 109°: This shape occurs when there is one lone pair and three bonded atoms around the central nitrogen atom. The bond angle is indeed close to 109°.

Based on the options provided and the common shapes of oxynitrogen ions, the correct answer would be:

- Trigonal planar, 120°

- Tetrahedral, 109.5°

- Bent, 120°

- Pyramidal, 109°

These shapes and bond angles are commonly observed in oxynitrogen ions depending on the arrangement of atoms and lone pairs around the central nitrogen atom.

Answer:

w=1.5t+12

Step-by-step explanation:

that's the right answer

The equation for the function W(t) representing the weight of the newborn baby t months after birth is W(t) = 16 + 1.5t

We can make use of the given data.

The infant gained weight at a pace of 1.5 pounds each month, reaching 16 pounds at the end of 4 months. With this knowledge, the equation can be written as follows:

W(t) = Initial weight + (Rate of weight gain per month) * (Number of months)

W(t) = 16 + (1.5 * t)

So, the equation for the function W(t) representing the weight of the newborn baby t months after birth is W(t) = 16 + 1.5t

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Answer:

k = +/- 6

Step-by-step explanation:

In the XY-Plane above, the circle has center (h,k) and radius 10. What is the value of k?

The concluding part of this question will be

Two coordinates are given, (4,0) and (20,0)

equation of a circle

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

We can then have two equations of the circle and solve simultaneously

([tex](4-h)^{2} +(-k)^{2} =10^{2}[/tex] .............................1

([tex](20-h)^{2} +(-k)^{2} =10^{2}[/tex] ......................................2

combining the two equation to solve

(20-h)^2 - (4-h)^2 = 0

20^2 - 40h +h^2 -4^2 + 8h -h^2 = 0

20^2 - 42 = 32h

h = 12

(4-12)^2 + (-k)^2 = 100

k^2 = 36

k = +/- 6

Answer:

  -273

Step-by-step explanation:

The n-th term of the sequence is given by ...

  an = a1 +d(n -1)

Filling in the given values and doing the arithmetic, we get ...

  a100 = 222 +(-5)(100 -1) = 222 -495

  a100 = -273

Answer:

-273

Step-by-step explanation:

Answer:the number of stamps that his cousin collected is 20. The variable is x which represents number of stamps. The equation is

4x = 60

Step-by-step explanation:

Let x represent the number of stamps that his cousin collected.

Xander collected four times as many stamps as his cousin. This means that the number of stamps that Xander collected is 4x

If xander collected 60 stamps, then it means that

4x = 60

x = 60/4 = 20 stamps

Final answer:

The number of stamps Xander's cousin collected is determined by dividing the number of stamps Xander collected, which is 60, by 4. Therefore, his cousin collected 15 stamps.

Explanation:

If Xander collected four times as many stamps as his cousin, and Xander collected 60 stamps, we can determine how many stamps his cousin collected by setting up an equation. Let's define c as the number of stamps Xander's cousin collected. According to the information provided, Xander collected four times this amount, so we have the equation 4c = 60. To find the value of c, we divide both sides of the equation by 4:

c = 60 / 4

c = 15

Therefore, Xander's cousin collected 15 stamps.

Answer:

the answer is na

Step-by-step explanation:

Final answer:

The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used with repetition.

Explanation:

The correct answer is A. The permutation rule (with different items) applies to this problem because repetition is allowed. The permutation rule (with some identical items) and the combination rule cannot be used with repetition. The calculations for this lottery game involve the permutation rule because the bettor selects four numbers between 0 and 9, and any selected number can be used more than once. The permutation rule allows for repetition and ensures that the order of the selected numbers is taken into account when determining the probability of winning the top prize.

Answer:

1) [tex]\Delta t=-0.5*t_{X}[/tex]

2) [tex]\Delta t=2*t_{X}[/tex]

Step-by-step explanation:

1) X=production rate of X (units/hour)

Y=production rate of Y (units/hour)

C=order (units)

t=working time (hour)

[tex](X+Y)*t_{toghether}=C[/tex]

[tex]X*t_{X}=C[/tex]

[tex]t_{toghether}=2/3*t_{X}[/tex]

Combining:

[tex](X+Y)*2/3*t_{X}=X*t_{X}[/tex]

[tex](X+Y)*2/3=X[/tex]

[tex]Y*2/3=X-2/3*X[/tex]

[tex]Y=2X[/tex]

In time: [tex]t_{Y}=0.5*t_{X}[/tex]

Difference of time: [tex]\Delta t=0.5*t_{X}-t_{X}=-0.5*t_{X}[/tex]

2) [tex]Y*t_{Y}=C[/tex]

[tex]X*t_{X}=C[/tex]

[tex]t_{Y}=2*t_{X}[/tex]

Difference of time: [tex]\Delta t=2*t_{X}-0.5*t_{X}=2*t_{X}[/tex]

Answer: 22

Step-by-step explanation:

The area of the picture =l×b = 20×16=320

The length of the frame is 16

If the width is x ( and it's uniform)

672- 320= 352

Area of the frame = 16× x=352

x=352/16

x = 22

Total width is 42 inch.

Given that,

According to the scenario, computation of the given data are as follows,

Total area of picture = 16 [tex]\times[/tex] 20 = 320 sq in.

So Frame area = 672 - 320 = 352

Now, length of frame = 16

Let, width of frame  =  w

So, 16 [tex]\times[/tex]  w = 352

W = 352 [tex]\div[/tex] 16

W = 22 inch

So total width is 22 + 20 = 42 inch.

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Answer:

Step-by-step explanation:

Reflection across the y-axis is a left-right reflection that changes only the sign of the x-coordinate. The transformation matrix (as for any rotation and/or reflection) can be 2-dimensional, and can left-multiply the coordinate matrix that has coordinate pairs as column vectors. The transformed coordinates show up as column vectors in the result matrix.

The only graphs showing left-right reflections are those of choices C and D. Only choice C has the points properly plotted on the graph.

The correct answer is graph C, or the third option.

Just got it right on edge 2020, hope this helps!! :)

To find the number of students who played soccer but not baseball or tennis, we need to analyze the given information and use the formula A = (A ∩ B) + (A ∩ C) + (A - A ∩ B ∩ C). Substituting the given values, we find that 32 students played soccer but not baseball or tennis.

To determine how many students played soccer but not baseball or tennis, we need to analyze the information given in the Venn diagram provided. Let's label the regions of the Venn diagram:

A represents the set of students who played soccer.

B represents the set of students who played baseball.

C represents the set of students who played tennis.

From the given information, we know that:

To find the number of students who played soccer but not baseball or tennis, we need to calculate the value of A without the intersection of the other sets. Using the formula:

A = (A ∩ B) + (A ∩ C) + (A - A ∩ B ∩ C),

we can substitute the given values:

A = 4 + 3 + (25 - 1)

A = 32

Therefore, 32 students played soccer but not baseball or tennis.

Answer:

7

Step-by-step explanation:

To find the maximum possible value of f(2), assume f(x) is a line with the greatest possible slope.  In this case, 5.

f(x) = 5x − 3

f(2) = 5(2) − 3

f(2) = 7

Answer:

B

Step-by-step explanation:

just swap around which one you multiply by 2 as they have multiplied the larger one by 2 instead of the smaller one.

Answer:

B.

Step-by-step explanation:

The error is on the first line:

4x + 20 = 2(3x + 5) is correct

- because AE = 2BD.

Answer:

Step-by-step explanation:

232/2=116

so 116,116 has maximum product

Answer:

The required function is F(x) = tanx - 2.

Step-by-step explanation:

We all know the graph of tanx ,

The graph of tanx passes through the origin and repeats in the interval of π.

Now , after looking into the graph we can clearly say that it is the shifted graph of tanx , Which is shifted in the -y direction .

Thus ,

It is clear that F(x) must be like F(x) = tanx - c , where c is a positive constant .

Also, At x = 0 , Value of the function is coming out to be -2 ,

Putting x = 0 in F(x) = tanx - c , we get , F(x) = -c .

Thus c = 2.

So, the required function is F(x) = tanx - 2.

Answer:

A. Between 14 and 15.

Step-by-step explanation:

Let x be the one leg of the right triangle.

We have been given that the legs of a right triangle are in the ratio of 3 to 1. So, the other leg of the right triangle would be 3x.

We are also told that the length of the hypotenuse of the triangle is √40.

Using Pythagoras theorem, we can set am equation as:

[tex]x^2+(3x)^2=(\sqrt{40})^2[/tex]

Let us solve for x.

[tex]x^2+9x^2=40[/tex]

[tex]10x^2=40[/tex]

[tex]\frac{10x^2}{10}=\frac{40}{10}[/tex]

[tex]x^2=4[/tex]

Take square root of both sides:

[tex]x=\sqrt{4}[/tex]

[tex]x=2[/tex]

The other leg would be [tex]3x\Rightarrow 3\cdot 2=6[/tex].

The perimeter of the triangle would be:

[tex]\text{Perimeter of triangle}=2+6+\sqrt{40}[/tex]

[tex]\text{Perimeter of triangle}=2+6+6.324555[/tex]

[tex]\text{Perimeter of triangle}=14.324555[/tex]

Therefore, the perimeter of the triangle is between 14 and 15 and option A is the correct choice.

Answer: the amount invested on the first mutual fund is $600 and the the amount invested on the second mutual fund is $1700

Step-by-step explanation:

Let x represent the amount of money that the investor invested in one mutual fund.

Let y represent the amount of money that the investor invested in the second mutual fund.

An investor invested a total of $2,300 in two mutual funds. This means that

x + y = 2300

Considering the first mutual fund, it earned a 8% profit. This means that amount earned is

8/100 × x = 0.08x

Considering the second mutual fund, it earned a 2% profit. This means that amount earned is

2/100 × y = 0.02y

If the investors total profit was $82, it means that

0.08x + 0.02y = 82 - - - - - - - - 1

Substituting x = 2300 - y into equation 1, it becomes

0.08(2300 - y) + 0.02y = 82

184 - 0.08y + 0.02y = 82

- 0.08y + 0.02y = 82 - 184

- 0.06y = - 102

y = - 102/- 0.06 = 1700

x = 2300 - y = 2300 - 1700

x = 600

Answer:

E. $926.24

Step-by-step explanation:

The total amount of interest paid is shown below:

1. For 6 months, the interest would be

= Invested amount × interest rate

= $10,000 × 2%

= $200

2. For 12 months, the interest would be

= (Invested amount + interest paid for 6 months)  × interest rate

= ($10,000 + $200) × 3%

= $10,200 × 3%

= $306

3. For 18 months, the interest would be

= (Invested amount + interest paid for 6 months + interest paid for 12 months )  × interest rate

=  ($10,000 + $200 + $306) × 4%

= $420

Now the total interest would be

= $200 + $306 + $420.24

= $926.24

Final answer:

The standard deviation for the given data is approximately 4.47.

Explanation:

The standard deviation is a measure of how spread out the data is from the mean. To calculate the standard deviation, we follow a formula that involves finding the difference between each data point and the mean, squaring those differences, and taking the average of the squared differences. Finally, we take the square root of the average to get the standard deviation.

For the given data, which is 13, 17, 9, and 21, we first find the mean (average) which is (13 + 17 + 9 + 21) / 4 = 60 / 4 = 15. Next, we calculate the squared differences from the mean for each data point:
(13 - 15)^2 = 4
(17 - 15)^2 = 4
(9 - 15)^2 = 36
(21 - 15)^2 = 36

Then, we find the average of the squared differences:
(4 + 4 + 36 + 36) / 4 = 80 / 4 = 20

Finally, we take the square root of the average:
sqrt(20) ≈ 4.47

So, the standard deviation for the given data is approximately 4.47.

Answer:

The height of the wall is approximately 10.17 feet.

Step-by-step explanation:

Given,

Length of wall = 30 feet

Total number of bricks = 2135

We have to find out the height of the wall.

To find out the height of wall, we have given the formula;

[tex]N=7LH[/tex]

Where N stands for number of bricks, L stands for length of wall and H stands for height of wall.

So we substitute the given values, we get;

[tex]2135=7\times30\times H\\\\2135=210H\\\\H=\frac{2135}{210}=10.166\approx10.17\ ft\\[/tex]

Hence the height of the wall is approximately 10.17 feet.

Answer:

3d

Step-by-step explanation:

A regular hexagon has six equivalent triangles, each triangle having one side of that of the hexagon. if the diameter of the circle in which the hexagon is inscribed is d , then its radius will be

[tex]r=\frac{d}{2}[/tex]

this radius also happens to be one of the side of the equivalent triangle whose one side is also the side of hexagon. since length of one side of hexagon is

[tex]\frac{d}{2}[/tex]

the circumference of the hexagon will be [tex]6 (\frac{d}{2} )\\\\thus \\circumference   \\of \\hexagon = 3d[/tex]

Answer:

  y = 2^x +5

Step-by-step explanation:

The number of new houses doubles each month, so the base (b) is that multiplier, 2. Then x is the number of months since the construction started (presumably at the end of the open house). If you want x to be something else, then h needs to be chosen so that (x -k) is the number of months the doubling has been taking place.

The value h represents the initial build of 5 houses. This is confirmed by the number 261 = 256 + 5 = 2^8 +5.

So, your equation for the number of houses built over time is ...

  y = 2^x +5

Answer: E. 120

The number of seniors were there in high school X at the beginning of the year = 120

Step-by-step explanation:

Given : At the beginning of the year, the ratio of juniors to seniors in high school X was 3 to 4.

Let the number of juniors be 3x and the number of seniors be 4x.

Since , during the year, 10 juniors and twice as many seniors transferred to another high school, while no new students joined high school X.

i.e. Number of juniors at the end of the year = 3x-10

Number of seniors at the end of the yea = 4x-2(10)=4x-20

At the end of the year, the ratio of juniors to seniors was 4 to 5.

[tex]\Rightarrow\dfrac{3x-10}{4x-20}=\dfrac{4}{5}[/tex]

[tex]\Rightarrow5(3x-10)=4(4x-20)[/tex]

[tex]\Rightarrow15x-50=16x-80[/tex]

[tex]\Rightarrow16x-15x=80-50[/tex]

[tex]\Rightarrow x=30[/tex]

The number of seniors were there in high school X at the beginning of the year = 4(30)=120

Hence, the correct answer is E. 120 .

To find the number of seniors at the high school X at the beginning of the year, calculate based on the given ratio and student transfers.

At the beginning of the year:

Answer:

Step-by-step explanation:

Wholesaler A is selling 10 pounds of chicken for $40.00. This means that the unit rate at which Wholesaler A is selling the chicken is

40/10 = $4 per pound of chicken.

Wholesaler B is selling 15 pounds of chicken for $45.00. This means that the unit rate at which Wholesaler B is selling the chicken is

45/15 = $3 per pound of chicken.

Wholesaler C is selling 20 pounds of chicken for $50.00. This means that the unit rate at which Wholesaler C is selling the chicken is

50/20 = $2.5 per pound of chicken.

Wholesaler C has the best price because she has the lowest unit rate pound of chicken.

Answer:

The statement is true only when both (1) and (2) are valid. If only one of (1) and (2) is valid, them the statement is not true.

Step-by-step explanation:

(1) alone is not sufficient, 27 is 3³ but is not a 12th power

(2) alone is not sufficient either, 81 is 3³ but it is not a 12th power

If both (1) and (2) are valid, then for each prime p that divides x, p should divide y and z, with y³ = x and z⁴=x.

Lets suppose that k is the highest power of p that divides y and m is the highest power that divides z, then (p^k)³ = (p^m)⁴. Therefore

p^3k = p^4m

This means that the power of p that appears on x is a multiple of both 3 and 4. Since those numbers are coprime, then that power is a multiple of 12.

This ensures that every prime dividing x has at least a power of 12 in the prime factirization, hence x is a 12th power.

Answer:

[tex]\large\boxed{1\dfrac{1}{3}\ u^2}[/tex]

Step-by-step explanation:

Let's sketch graphs of functions f(x) and g(x) on one coordinate system (attachment).

Let's calculate the common points:

[tex]x^2=\sqrt{x}\qquad\text{square of both sides}\\\\(x^2)^2=\left(\sqrt{x}\right)^2\\\\x^4=x\qquad\text{subtract}\ x\ \text{from both sides}\\\\x^4-x=0\qquad\text{distribute}\\\\x(x^3-1)=0\iff x=0\ \vee\ x^3-1=0\\\\x^3-1=0\qquad\text{add 1 to both sides}\\\\x^3=1\to x=\sqrt[3]1\to x=1[/tex]

The area to be calculated is the area in the interval [0, 1] bounded by the graph g(x) and the axis x minus the area bounded by the graph f(x) and the axis x.

We have integrals:

[tex]\int\limits_{0}^1(\sqrt{x})dx-\int\limits_{0}^1(x^2)dx=(*)\\\\\int(\sqrt{x})dx=\int\left(x^\frac{1}{2}\right)dx=\dfrac{2}{3}x^\frac{3}{2}=\dfrac{2x\sqrt{x}}{3}\\\\\int(x^2)dx=\dfrac{1}{3}x^3\\\\(*)=\left(\dfrac{2x\sqrt{x}}{2}\right]^1_0-\left(\dfrac{1}{3}x^3\right]^1_0=\dfrac{2(1)\sqrt{1}}{2}-\dfrac{2(0)\sqrt{0}}{2}-\left(\dfrac{1}{3}(1)^3-\dfrac{1}{3}(0)^3\right)\\\\=\dfrac{2(1)(1)}{2}-\dfrac{2(0)(0)}{2}-\dfrac{1}{3}(1)}+\dfrac{1}{3}(0)=2-0-\dfrac{1}{3}+0=1\dfrac{1}{3}[/tex]

The area bounded by the functions f(x) and g(x) in graph below.

The given function are f(x)=x² and g(x)=√x.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions.

Area bounded = |x²-√x|

Find the domain by finding where the expression is defined.

Interval Notation:

[0,∞)

Set-Builder Notation:{x|x≥0}

Therefore, the area bounded by the functions f(x) and g(x) in graph below.

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The cost of one pack of coffee is $12 and cost of one tea pack is $6.

Step-by-step explanation:

Let,

Cost of one pack of coffee = x

Cost of one pack of tea = y

According to given statement;

39x+45y=738    Eqn 1

11x+13y=210       Eqn 2

Multiplying Eqn 1 by 11

[tex]11(39x+45y=738)\\429x+495y=8118\ \ \ Eqn\ 3\\[/tex]

Multiplying Eqb 2 by 39

[tex]39(11x+13y=210)\\429x+507y=8190\ \ \ Eqn\ 4[/tex]

Subtracting Eqn 3 from Eqn 4

[tex](429x+507y)-(429x+495y)=8190-8118\\429x+507y-429x-495y=72\\12y=72[/tex]

Dividing both sides by 12

[tex]\frac{12y}{12}=\frac{72}{12}\\y=6[/tex]

Putting y=6 in Eqn 2

[tex]11x+13(6)=210\\11x+78=210\\11x=210-78\\11x=132[/tex]

Dividing both sides by 11

[tex]\frac{11x}{11}=\frac{132}{11}\\x=12[/tex]

The cost of one pack of coffee is $12 and cost of one tea pack is $6.

Keywords: linear equation, subtraction

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Answer:

A pack of coffee=$12

A pack of tea=$6

Step-by-step explanation:

Let the cost of 1 pack of coffee =c

and the cost of 1 pack of tea =t

we can the write the following equations

39c+45t=738...........eqn(1)

11c+13t=738...........eqn(2)

Multiply eqn(1) by 11

429c+495t=8118...........eqn(3)

Multiply eqn(2) by 39

429c+507t=8190...........eqn(4)

eqn(4)-eqn(3)

This implies

12t=72

Dividing through by 12 ,we get

t=6

substituting the value of t into equation two,we obtain

11c+13(6)=210

11c+78=210

11c=210-78

11c=132

Dividing through by 11,we obtain

c=12