The area of a trapezoid is 140 cm2 and the height is 20 cm. What is the sum of the bases?

A. 14 cm

B. 49 cm

C. 60 cm

D. 7 cm

Answer:

Step-by-step explanation:

The progression of sums is ...

  1, 5, 15, 34, 65, ...

So, first differences are ...

  4, 10, 19, 31

Second differences are ...

  6, 9, 12, ...

Third differences are constant:

  3, 3, ...

This means the expression for Sn will be a cubic expression. If dn is the first of the n-th differences, then the equation can be written as ...

  Sn = S1 +(n -1)(d1 +(n -2)/2(d2 +(n -3)/3(d3)))

And this simplifies a little bit to ...

  Sn = 1 +(n -1)(4 +(n -2)(n +3)/2)

In simpler form, we have ...

  Sn = 1/2(n³ +n)

Then the two terms we're interested in are ...

  S7 = (1/2)(7³ +7) = 175

  S17 = (1/2)(17³ +17) = 2465

Each term Sₙ consists of the sum of a triangular number of terms, which are given by

[tex]T_n = \displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]

The triangular numbers are given recursively for n ≥ 1 by

[tex]T_n = T_{n-1} + n[/tex]

starting with T₀ = 0.

For example,

• S₁ = 1 and

[tex]\displaystyle S_1 = \sum_{k=T_0+1}^{T_1} k = \sum_{k=1}^1 k = 1[/tex]

• S₂ = 2 + 3 and

[tex]\displaystyle S_2 = \sum_{k=T_1+1}^{T_2} k = \sum_{k=2}^3 k = 2 + 3[/tex]

• S₃ = 4 + 5 + 6 and

[tex]\displaystyle S_3 = \sum_{k=T_2+1}^{T_3} k = \sum_{k=4}^6 k = 4 + 5 + 6[/tex]

Then the n-th term of the sequence we're considering is

[tex]S_n = \displaystyle \sum_{k=T_{n-1}+1}^{T_n} k = \sum_{k=T_{n-1}+1}^{T_{n-1}+n} k[/tex]

Expanding this sum, we have

[tex]S_n = \left(T_{n-1}+1\right) + \left(T_{n-1}+2\right) + \left(T_{n-1}+3\right) + \cdots + \left(T_{n-1}+n\right)[/tex]

There are n terms on the right side, and hence n copies of [tex]T_{n-1}[/tex], and the rest of the terms make up the next triangular number [tex]T_n[/tex] :

[tex]S_n = nT_{n-1} + 1 + 2 + 3 + \cdots + n[/tex]

[tex]S_n = nT_{n-1} + \displaystyle \sum_{k=1}^n k[/tex]

[tex]S_n = nT_{n-1} + T_n[/tex]

We have a closed form for [tex]T_n[/tex], so we end up with

[tex]S_n = n \cdot \dfrac{(n-1)n}2 + \dfrac{n(n+1)}2 \implies \boxed{S_n=\dfrac{n^3+n}2}[/tex]

From here it's easy to find S₇ and S₁₇.

[tex]S_7 = \dfrac{7^3+7}2 \implies \boxed{S_7 = 175}[/tex]

[tex]S_{17} = \dfrac{17^3+17}2 \implies \boxed{S_{17} = 2465}[/tex]

Answer:

x=13

Step-by-step explanation:

8x-20=4x+32

start by moving the terms,

8x-4x=32+20

combine

4x=52

divide each side by 4

x=13

Answer:

48 is 25% of 12

Step-by-step explanation:

Answer:

The line shifted up 8 units

Step-by-step explanation:

Answer:

it is d

Step-by-step explanation:

Answer:

It is B

Step-by-step explanation:

Answer:

25 and 57.

Step-by-step explanation:

Let the lowest number be x then the other is 2x + 7, So:

x(2x + 7) = 1425

2x^2 + 7x - 1425 = 0

2 * - 1425 = -2850 = -2 * 3 * 5 * 5 * 19

We need 2 numbers whose product is -2850 and whose sum is + 7.

These are 3 * 19 = 57 and -2* 5 * 5 = -50 so we write:

2x^2 - 50x + 57x - 1425 = 0  Now we factor by grouping:

2x(x - 25) + 57( x - 25) = 0)

(2x + 57) ( x - 25) = 0

x = -57, 25.

We ignore the negative as we are given that the number is positive,

x = 25 is one number and the other is 2(25) + 7 = 57.

Answer:

The answer is 9

Explanation:

0.5a+8.75=13.25

We simplify the equation to the form, which is simple to understand

0.5a+8.75=13.25

We move all terms containing a to the left and all other terms to the right.

+ 0.5a=+13.25-8.75

We simplify left and right side of the equation.

+ 0.5a=+4.5

We divide both sides of the equation by 0.5 to get a.

a=9

Answer:

See answer below.

Step-by-step explanation:

This is true for all right triangle a² + b² = c²    were c = the largest number

9, 12, 15           A right triangle

21, 28, 35       A right triangle

9, 13, 16          Not a right triangle

10, 11, 15          Not a right triangle

Answer:

The choice 66

Step-by-step explanation:

[tex]6x + 2m \\ 6(14) + 2( - 9) \\ 84 - 18 \\ = 66[/tex]

Answer:

What supplies?

Step-by-step explanation:

Answer:

I think I've answered this question already.

Step-by-step explanation:

Answer:

[tex]f^{-1} (x) =x-11[/tex]

Step-by-step explanation:

[tex]f(x) = x+11\\\\\text{Write f(x) as y = x+11}\\\\\text{Replace x with y,}\\\\x = y+11\\\\\text{Solve for y}\\\\x=y+11\\\\\implies y= x-11\\\\\implies f^{-1} (x) = x-11[/tex]

Answer:

A. 30% B. 45%

Step-by-step explanation:

A. 72,000/240,000 = x/100

(72,000)(100) = (240,000)x

7,200,000 = 240,000x

30 = x

B. 108,000/240,000 = x/100

10,800,000 = 240,000x

45 = x

Answer: 4.5

Step-by-step explanation:

30% is equal to 30/100, which is 0.3

If you times 0.3 by 15, you will get 30% of 15

Therefore the answer would be 0.3 x 15 = 4.5

Work Shown:

1.45*270 = 391.50

The multiplier 1.45 represents a 45% increase

Think of it as 1.45 = 1+0.45 = 100% + 45%

Answer:

391.5

Step-by-step explanation:

the 45% of $270 is $121.5

The distance Jen is from the base of the lighthouse is 86.6 meters.

Trigonometric ratio is used to show the relationship between the sides of a and angles of a right angled triangle.

Let d represent the distance Jen is from the base of the lighthouse.

Using trigonometric ratios:

[tex]tan(30)=\frac{50}{d} \\\\d=\frac{50}{tan(30)} \\\\d=86.6\ m[/tex]

The distance Jen is from the base of the lighthouse is 86.6 meters.

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Step-by-step explanation:

in total we have 3+4 = 7 balls.

when we draw the first ball, the probability to draw a red ball is 3/7, and a blue ball 4/7.

when we draw the second ball, we have now only 6 balls in total.

the probabilty to draw a red back now depends also on the result of the first draw.

if the first ball was already red, then we have only a chance now of 2 out of 6.

if the first ball was blue, then we have now a chance of 3 out of 6.

so, the probably to draw at least 1 red ball in 2 draws is the probability of drawing one on the first draw plus the probability of drawing one on the second :

1 - probability to see 2 blue balls

1 - 4/7 × 3/6 = 1 - 12/42 = 30/42 = 0.714285714...

the expected number of red balls in 2 draws is

1 red in first red in first red in second

but not second and second but not first

1×(3/7 × 4/6) + 2×(3/7 × 2/6) + 1×(4/7 × 3/6) = 12/42 + 12/42 + 12/42 = 36/42 = 6/7 = 0.857142857 ≈ 0.8571

Answer:

Its 78,300

Step-by-step explanation:

m=-1 and c=8 for the line  y = 8 − x

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

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

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

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

The given equation is  y = 8 − x

This equation we can write in the form of slope intercept form

y=-x+8

y=(-1)x+8

Now compare the above equation with slope intercept form

m=-1 and c=8

Hence, m=-1 and c=8 for the line  y = 8 − x

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Answer:

The price of 1 adult ticket is 12 dollars, and the price of a ticket for one student is 7 dollars

Step-by-step explanation:

Make a system of equations for the two days that the play was shown.

Let x = the price of an adult ticket

Let y = the price of a student ticket

For the first day:

For the second day:

Now, we can solve using the elimination method. Multiply the first equation by 2 and the second equation by 9. Then swap the order of the equations.

Answer: And I don’t know math

Step-by-step explanation:

Answer:

21x-28

Step-by-step explanation:

the distributive property states that a(bx+c) is equal to abx+ac. in this expression, a is 7, b is 3, and c is negative 4. ab is 7*3, which is 21, so abx is 21x. ac is 7*-4, which is -28. combining these two results in 21x-28.

Answer:

21x-28

Step-by-step explanation:

distribute the 7 to the 3x & the -4

you get 21x-28

Answer:

[tex] \frac{1}{2} \: \: \: \: \: \frac{1}{4 \: } \: \: \: \frac{1}{8} [/tex]

[tex] \frac{4}{8} \: \: \: \frac{2}{8} \: \: \: \frac{1}{8} [/tex]

She ate

[tex] \frac{7}{8} [/tex]

[tex]slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= 2x^3+4\qquad \begin{cases} x_1=4\\ x_2=6 \end{cases}\implies \cfrac{f(6)-f(4)}{6-4} \\\\\\ \cfrac{[2(6)^3+4]~~ -~~[2(4)^3+4]}{2}\implies \cfrac{436~~ -~~132}{2}\implies \cfrac{304}{2}\implies 152[/tex]

Step-by-step explanation:

An acute triangle is a triangle whose angles are all acute (i.e. less than 90°). In the acute triangle shown below, ∠a, ∠b and ∠c are all acute angles. Example 1: A triangle has angles 46°, 63° and 71°.

Step-by-step explanation:

Relation

Hope it correct!!!

The vertices of the feasible region of the constraints is (0,2.5)

The constraints are given as:

x + 3y ≤ 6

4x + 6y ≥ 9

x ≥ 0,  y ≥ 0

Express the inequalities as equations

x + 3y = 6

4x + 6y = 9

Make x the subject in x + 3y = 6

x = 6 - 3y

Substitute x = 6 - 3y in 4x + 6y = 9

4(6 - 3y) + 6y = 9

Expand

24 - 12y + 6y = 9

Evaluate the like terms

-6y = -15

Divide by 6

y = 2.5

Substitute y = 2.5 in x = 6 - 3y

x = 6 - 3 * 2.5

Evaluate

x = -1.5

So, we have:

(x,y) = (-1.5, 2.5)

Recall that: x ≥ 0,  y ≥ 0

This means that the feasible region must have positive coordinates or zero

So, we set x = 0

(x,y) = (0, 2.5)

Hence, the vertices of the feasible region of the constraints is (0,2.5)

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