Write the augmented matrix of the system of equations given below. (10 points)
−2 + 2 − = −13
−3 − + = 7
− + 3 − 2 = 0

Answer:

The answer is ninth term is -32

Answer: 4.8

Step-by-step explanation:

Corresponding sides of similar figures are proportional, so

[tex]\frac{x}[4}=\frac{6}[5}\\\\x=4\left(\frac{6}{5} \right)=\boxed{4.8}[/tex]

(a.i) If [tex]a_1,a_2,a_3,\ldots[/tex] are in arithmetic progression, then there is a constant [tex]d[/tex] such that

[tex]a_{n+1} - a_n = d[/tex]

for all [tex]n\ge1[/tex]. In other words, the difference [tex]d[/tex] between any two consecutive terms in the sequence is always the same.

[tex]a_2-a_1 = a_3-a_2 = a_4-a_3 = \cdots = d[/tex]

Now, we can expand the target expression into partial fractions.

[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha}{a_n} + \dfrac{\beta}{a_{n+1}}[/tex]

Combining the fractions on the right and using the recursive equation above, we have

[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha (a_n + d) + \beta a_n}{a_n(a_n+d)} = \dfrac{(\alpha+\beta) a_n + \alpha d}{a_n a_{n+d}} \\\\ \implies \begin{cases}\alpha + \beta = 0 \\ \alpha d = 1 \end{cases} \implies \alpha = \dfrac1d, \beta = -\dfrac1d[/tex]

and hence

[tex]\dfrac1{a_n a_{n+1}} = \dfrac1{da_n} - \dfrac1{da_{n+1}}[/tex]

as required.

(a.ii) Using the previous result, the [tex]n[/tex]-th term [tex](n\ge1)[/tex] in the sum on the left is

[tex]\dfrac1{a_n a_{n+1}} = \dfrac1d \left(\dfrac1{a_n} - \dfrac1{a_{n+1}}\right)[/tex]

Expand each term in this way to reveal a telescoping sum:

[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_{99}a_{100}} \\\\ ~~~~~~~~ = \dfrac1d \left(\left(\dfrac1{a_1} - \dfrac1{a_2}\right) + \left(\dfrac1{a_2} - \dfrac1{a_3}\right) + \left(\dfrac1{a_3} - \dfrac1{a_4}\right) + \cdots + \left(\dfrac1{a_{99}} - \dfrac1{a_{100}}\right)\right) \\\\ ~~~~~~~~ = \dfrac1d \left(\dfrac1{a_1} - \dfrac1{a_{100}}\right) = \dfrac{a_{100} - a_1}{d a_1 a_{100}}[/tex]

By substitution, we can show

[tex]a_n = a_{n-1} + d = a_{n-2} + 2d = \cdots = a_1 + (n-1)d \\\\ \implies a_{100} = a_1 + 99d[/tex]

so that the last expression reduces to

[tex]\dfrac{(a_1 + 99d) - a_1}{d a_1 a_{100}} = \dfrac{99d}{d a_1 a_{100}} = \dfrac{99}{a_1 a_{100}}[/tex]

as required. More generally, it's easy to see that

[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac{n}{a_1a_{n+1}}[/tex]

(b) I assume you mean the equation

[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{k(k+4)} = \dfrac2{25}[/tex]

Note that the distinct factors of each denominator on the left form an arithmetic sequence.

[tex]a_1 = 3[/tex]

[tex]a_2 = 3 + 4 = 7[/tex]

[tex]a_3 = 7 + 4 = 11[/tex]

and so on, with [tex]n[/tex]-th term

[tex]a_n = 3 + (n-1)\times4 = 4n - 1[/tex]

Let [tex]a_n=k[/tex]. Using the previous general result, the left side reduces to

[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac n{3a_{n+1}} \\\\ \implies \dfrac{\frac{k+1}4}{3(k+4)} = \dfrac2{25}[/tex]

Solve for [tex]k[/tex].

[tex]\dfrac{k+1}{12k+48} = \dfrac2{25} \implies 25(k+1) = 2(12k+48) \\\\ \implies 25k + 25 = 24k + 96 \implies \boxed{k=71}[/tex]

The distance from ships to the submarine is AX=1084.20

                                                                      BX=1270.69

Let X be the submarine position.

The length between a and b is AB=1425

             and the angle point a is at 59 degrees

                    the angle point b is at 47 degrees

The calculating angle at X:

∠X+∠A+∠B = 180

∠X+59°+47°=180°

∠X=180°-59°-47°

∠X=74°

Then the distance between boat A and the submarine will be found using sine law

Sine law is the ratio of each side of a plane triangle to the sine of the opposite angle is the same for all three sides and angles.

[tex]\frac{a}{sinA}=\frac{b}{sin B} = \frac{c}{sin C}[/tex]

so for Boat A and submarine, we use

[tex]\frac{AB}{sin X}=\frac{AX}{sin B}[/tex]

[tex]\frac{1425}{sin (74)} = \frac{AX}{sin (47)}[/tex]

When AX is the subject

AX=[tex]\frac{1425}{sin(74)}*sin(47)[/tex]

=[tex]\frac{1425}{0.9613}*0.7314[/tex]

AX=1042.245/0.9613

AX=1084.20

The distance between ship B and the submarine

The sine formula we use is

[tex]\frac{AB}{sin X}=\frac{BX}{sin A}[/tex]

[tex]\frac{1425}{sin 74}=\frac{BX}{sin 59}[/tex]

When BX is the subject

BX=[tex]\frac{1425}{sin 74}*sin(59)[/tex]

= 1425/0.9613 * 0.8572

=1221.51/0.9613

=1270.69

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Answer:

Ships A and B are 1,425 feet apart and detect a submarine below them. The angle of depression from ship A to the submarine is 59°, and the angle of depression from ship B to the submarine is 47°.

How far away is the submarine from the two ships? Round to the nearest hundredth of a foot.

The distance from ship A to the submarine is about

✔ 1,084.18

feet.

The distance from ship B to the submarine is about

✔ 1,270.69

feet.

Step-by-step explanation:

[tex]frequency = \frac{24 \: osc}{60 \: sec} = 0.4 \: osc \: per \: sec[/tex]

[tex]period = \frac{1}{frequency} = \frac{1}{0.4} = 2.5 \: sec \: per \: osc[/tex]

[tex]t = 2\pi \sqrt{ \frac{l}{g} } \\ 2.5 = 2\pi \sqrt{ \frac{l}{9.8} } \\ \frac{2.5}{2\pi} = \sqrt{ \frac{l}{9.8} } \\ ( \frac{2.5}{2\pi} ) {}^{2} = \frac{l}{9.8}[/tex]

[tex]l = ( \frac{2.5}{2\pi} ) {}^{2} \times 9.8 = 1.55148 \: meters[/tex]

The trigonometric function gives the ratio of different sides of a right-angle triangle. The measure of ∠C is 28°. And the value of b and c is 7.06 and 3.76, respectively.

The trigonometric function gives the ratio of different sides of a right-angle triangle.

[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]

where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite the 90° angle.

The sum of all the angles can be written as,

∠A + ∠B + ∠C = 180°

90° + 62° + ∠C = 180°

∠C = 28°

Sin(∠B) = AC/BC

Sin(∠B) = b/8

b = 8 × Sin(62°)

b = 7.06

Cos(∠B) = AB/BC

Cos(∠B) = c/8

c = 8 × Cos(62°)

c = 3.76

Hence, the measure of ∠C is 28°. And the value of b and c is 7.06 and 3.76, respectively.

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Answer:

  B.  (5x+5)/(x²-2x)

Step-by-step explanation:

As with numerical fractions, division by a rational expression is equivalent to multiplication by its reciprocal.

  [tex]\dfrac{5}{x-2}\div\dfrac{x}{x+1}=\dfrac{5}{x-2}\times\dfrac{x+1}{x}[/tex]

As with numerical fractions, the product is the product of numerators, divided by the product of denominators.

  [tex]=\dfrac{5(x+1)}{x(x-2)}=\boxed{\dfrac{5x+5}{x^2-2x}}[/tex]

Answer: C

Step-by-step explanation:

The line is dotted, so eliminate B and D.

Also, the line has a y-intercept of 1, so this eliminates A.

This means the answer must be C.

Answer: C. 0.0225

Step-by-step explanation:

The distance is 0.03 - 0.02 = 0.01.

1/4 of this is 0.0025.

Adding this to 0.02, we get 0.00225.

Volume is a three-dimensional scalar quantity. The length of the square base side is 5 meters.

A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.

The volume of the square pyramid is one-third of the product of the base area and the height. Therefore, the area of the base of the pyramid will be,

The volume of pyramid = (1/3) × Height × (Base length)²

75 m³ = (1/3) × 9 × (Base length)²

Base length)² = 25 m²

Base length = 5 m

Hence, the length of the square base side is 5 meters.

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The length of the arc in the image given is approximately: A. 3.5 cm

Length of arc = ∅/360 × 2πr

The missing parts of the question is shown in the image below, where we are asked to find arc length of JH.

Thus, we would have the following:

∅ = 25°

Radius (r) = 16/2 = 8 cm

Substitute the values into the formula:

Arc length JH = 25/360 × 2π(8)

Arc length JH ≈ 3.5 cm

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The value of the hypotenuse is C = 11.73

We can see that C is the hypotenuse of the right triangle.

We can see that the top angle measures 43°, and the opposite cathetus to that angle measures 8 units.

Then we can use the rule:

Sin(a) = (opposite cathetus)/(hypotenuse).

Replacing what we know:

Sin(43°) = 8/C

Solving for C:

C = 8/Sin(43°) = 11.73

The value of the hypotenuse is C = 11.73

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Answer:

no

Step-by-step explanation:

Prime factorization of 66

[tex]66=[/tex][tex]2[/tex] × [tex]3[/tex] × [tex]11[/tex]

the perfect cubes are the numbers that possess exact cubic roots

[tex]\sqrt[3]{66}[/tex] ≈ [tex]4.04[/tex]

66 is not a perfect cube

Hope this helps

Answer:

[tex]2.20b + 3.60m = 44.20\\2.50b + 3.40m = 44.70[/tex]

Step-by-step explanation:

So in this systems of equations, we know the prices of the items, so the coefficients will represent how many Mary is getting of that item. So you can represent amount of bread as the variable b and the amount of milk as the variable m. On the right will be the total amount spent, since multiplying the price by how much you buy should get you the price

So since at store A, she spends 44.20 and bread costs 2.20 and milk costs 3.60 you get

[tex]2.20b + 3.60m = 44.20[/tex]

Since at store B, she spends 44.70 and bread costs 2.50 and milk costs 3.40 you get the equation:

[tex]2.50b + 3.40m = 44.70[/tex]

Answer:

Option A

2.20b + 3.6m = 44.20

2.5b + 3.4m = 44.70

Explanation:

For store A, bread costs $2.20 and milk costs $3.60 and she pays $44.20

Equation created: 2.20b + 3.6m = 44.20

For store B, bread costs $2.50 and milk costs $3.40 and she pays $44.70

Equation created: 2.5b + 3.4m = 44.70

Answer:

C. [tex]1\frac{1}{2} x - 1\frac{3}{4}[/tex]

Step-by-step explanation:

c

Answer:

[tex]1\frac{1}{2}x - 1\frac{3}{4}[/tex]

Step-by-step explanation:

So the first step is to distribute the [tex]\frac{1}{4}[/tex] to the (3x - 7) this gives you:

[tex]\frac{3}{4}x-\frac{7}{4} + \frac{3}{4}x[/tex]

Add like terms:

[tex]\frac{6}{4}x-\frac{7}{4}[/tex]

Simplify the fractions:

[tex]\frac{3}{2}x-\frac{7}{4}[/tex]

Convert it to a mixed fraction:

[tex]1\frac{1}{2}x - 1\frac{3}{4}[/tex]

Answer:

Step-by-step explanation:

A bit late, but better than never.

24. For [tex]x\neq1[/tex], we have

[tex]\dfrac{9x^2 - x - 8}{x - 1} = \dfrac{(x-1)(9x+8)}{x-1} = 9x+8[/tex]

Then as [tex]x[/tex] approaches 1, we have

[tex]\displaystyle \lim_{x\to1} f(x) = \lim_{x\to1} (9x+8) = 17 \neq 0[/tex]

so the function is not continuous at [tex]x=1[/tex]. It is thus continuous on the interval union [tex](-\infty,0)\cup(0,\infty)[/tex]. You can also write this as "[tex]x<0[/tex] or [tex]x>0[/tex]".

25. When [tex]x[/tex] approaches 2 from the left (when [tex]x<2[/tex]), we have

[tex]\displaystyle \lim_{x\to2^-} f(x) = \lim_{x\to2} (5-x) = 3[/tex]

When [tex]x[/tex] approaches 2 from the right [tex](x>2)[/tex], we have

[tex]\displaystyle \lim_{x\to2^+} f(x) = \lim_{x\to2} (2x-3) = 1[/tex]

so the function is not continuous at [tex]x=2[/tex]. Thus it's continuous on [tex](-\infty,2)\cup(2,\infty)[/tex], or "[tex]x<2[/tex] or [tex]x>2[/tex]".

Answer:

no because 1 x value cannot have 2 y values , i hope the helps

Choose C.

Step-by-step explanation:

the distance between 2 points is per Pythagoras

c² = a² + b²

with c being the Hypotenuse (the baseline opposite of the 90° angle, which is the direct distance between the points). a and b are the legs (the x and y coordinate differences).

so,

AB² = (3-6)² + (1-2)² = 9+1 = 10

AB = sqrt(10) = 3.16227766...

BC² = (6-6)² + (2 - -2)² = 0² + 4² = 16

BC = sqrt(16) = 4

CD² = (6-3)² + (-2 - -3)² = 3² + 1² = 10

CD = sqrt(10) = 3.16227766...

DA² = (3-3)² + (-3 - 1)² = 0² + 4² = 16

DA = sqrt(16) = 4

so, the perimeter is

2×4 + 2×sqrt(10) = 8 + 6.32455532... = 14.32455532... ≈

≈ 14.3

Answer:

2¹²

Step-by-step explanation:

one expression equivalent to 16³ is 2¹²

think about it like this: we know that 2 to the power of 4 is 16

(2 x 2 = 4, x 2 = 8, x 2 = 16)

So for every "16" we already have 2⁴; meaning that 16³ is the same thing as 3 groups of 2⁴ -- which is 2¹²

hope this helps! have a lovely day :)

The value of z in the equation evaluation as given in the task content is; 3/5.

It follows from the task content that the value of z as in the task content can be evaluated as follows;

2 4/15-z= 1 2/3

34/15 - z = 5/3

Hence, z = 34/15 - 5/3

z = (34-25)/15 = 9/15

z = 3/5

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It will take the roller coaster 171.652 seconds to pass over the hill and reach ground level

The function is given as:

h = –.025t^2 + 4t + 50

Set h = 0.

So, we have:

–.025t^2 + 4t + 50 = 0

Using a graphing calculator, we have:

t = -11.652 and t = 171.652

Time cannot be negative.

So, we have:

t = 171.652

Hence, it will take the roller coaster 171.652 seconds to pass over the hill and reach ground level

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Step-by-step explanation:

f.g(x) = f(x+2)

         =(x+2)2 -1

         =(x2+2*x*2+4)-1

         =x2+4x+4-1

         =x2+4x+3

Answer:1,3,5

Step-by-step explanation:

The value of F(G(x)) is 2x^2 - 13

The functions are given as:

F(x)=2x+1

G(x)=x^2-7

To find (FxG)(x), we make use of

(FxG)(x) = F(G(x))

So, we have:

F(G(x))=2G(x)+1

Substitute G(x)=x^2-7

F(G(x))=2(x^2-7)+1

Expand

F(G(x))=2x^2 - 14 + 1

Evaluate the like terms

F(G(x))=2x^2 - 13

Hence, the value of F(G(x)) is 2x^2 - 13

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Answer:

Twenty - one miles per gallon

Step-by-step explanation:

⇒ 550/26 = Amount per gallon which is one

⇒21.15384615384615

[tex] \approx [/tex] 21.2 miles per gallon

The distance travelled by truck in one gallon of fuel is 21.2 miles per gallon.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that a truck can travel 550 miles on a tank of gasoline. The distance travelled by truck will be calculated as,

550 miles = 26 gallons

1 gallon = 550 / 26

1 gallon = 21.2 miles per gallon

Therefore, the distance travelled by truck in one gallon of fuel is 21.2 miles per gallon.

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Answer:

AB: y=2x+2

CB: y=-2x+6

CD: y=1/2x-1.5

DA: y=-3x+2

Step-by-step explanation:

Answer:

AB: [tex]y=2x+2[/tex]

CB: [tex]y=-2x+6[/tex]

CD: [tex]y=\frac{1}{2}x -1.5[/tex]

DA: [tex]y=-3x+2[/tex]

Step-by-step explanation:

So when it's saying y = __ x + __ it's asking for the slope-intercept form. This is given as y=mx+b where m is the slope and b is the y-intercept. m is the slope because as x increases by 1, the y-value will increase by m, which is by definition what the slope is [tex]\frac{rise}{run}[/tex] how much the function "rises" as x "runs".

AB: See how it "rises" by 2 and only "runs by 1", thus the slope is 2/1 or 2. the last part is the y-intercept, which is the value of y when the line crosses the y-axis. If you look at the graph when it "crosses" the y-axis it's y-value is 2. So you have the equation: [tex]y=2x+2[/tex]

CB: See how it "goes down" by 4 and "runs" by 2, thus the slope is -4/2 or -2. The y-intercept isn't shown on the graph but you can calculate that by substituting known values into the slope-intercept form. So we know so far y=-2x+b since the slope was calculated, and we can take any point on the line to calculate b, for this example I'll take the point C which is (3, 0) which is (x, y) and I'll plug that in

Plug values in:

0 = -2(3) + b

0 = -6 + b

6 = b

This gives you the complete equation: [tex]y=-2x+6[/tex]

CD: see how it only "rises" by 1 but "runs" by 2, thus the slope is 1/2. The y-intercept isn't shown on the graph but you can calculate it by substituting a point into the equation For this example I'll use point C (3, 0)

0 = 3(1/2) + b

0 = 1.5 + b

-1.5 = b

This gives you the complete equation: [tex]y=\frac{1}{2}x - 1.5[/tex]

DA: see how it "decreases" by 3 but "runs" by 1, thus the slope is -3/1 or -3. The y-intercept is known and is at (0, 2) so now we plug these values in to get the equation: [tex]y=-3x+2[/tex]

Answer:

D

Step-by-step explanation:

Subtracting 2 from both sides and then dividing both sides by 3 gives that x² = -2/3, which indicates the roots are complex.