All Answers

#2

+33661

+1

(1/2) * 11 * h = (1/2) * (h - 5) * 16 ?????!!!!!

AlanOct 12, 2023

#2

+4

+2

Thanks for helping me

CoolMathUser123Oct 12, 2023

#1

+789

-2

We can start by factoring 125:

125 = 5^3

Therefore, we can rewrite the equation as follows:

5^3x = a√b

We can also rewrite the right-hand side of the equation as follows:

a√b = a^(1/2)b^(1/4)

Setting the powers of 5 equal on both sides of the equation, we get:

3 = a^(1/2)b^(1/4)

Squaring both sides of the equation, we get:

9 = ab^(1/2)

Therefore, a=9 and b=16.

In conclusion, a=9 and b=16.

bingboyOct 12, 2023

#1

+789

-1

We can factor the left-hand side of the equation as follows:

(3x - 27)^3 + (27x - 3)^3 = ((3x - 27) + (27x - 3))(((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2)

= (30x - 30)((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2).

We can factor the right-hand side of the equation as follows:

(3x + 27x - 30)^3 = (60x - 30)^3.

Since (60x−30)3=(30x−30)((60x−30)2), we can write the equation as follows:

(30x - 30)((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2) = (60x - 30)((60x - 30)^2).

Since (30x−30)=0, we can cancel it from both sides of the equation, which gives us:

((3x - 27)^2 - (3x - 27)(27x - 3) + (27x - 3)^2) = (60x - 30)^2.

Expanding both sides of the equation, we get:

9x^2 - 162x + 729 - 27x^3 + 196x^2 - 171x + 243 + 729x^2 - 189x + 9 = 3600x^2 - 7200x + 2700

Combining like terms, we get:

2547x^2 - 2316x - 2507 = 0

This quadratic factors as follows:

(x - 1)(2547x + 2507) = 0

The only real solution to this equation is x=1

bingboyOct 12, 2023

#1

+789

0

First, we can factor out a 21 from all the terms:

(a + b)/2 - a/2 - b/2 = \frac{1}{2} (a + b - a - b)

Then, we can combine the a and b terms:

\frac{1}{2} (a + b - a - b) = \frac{1}{2} \cdot 0 = \boxed{0}

bingboyOct 12, 2023

#1

+789

-2

The expression (kn)+(n−kn) is always equal to (kn+1), which can be proven using Pascal's identity. Therefore, \begin{align*} \binom{3}{1} + \binom{3}{2} &= \binom{3 + 1}{1} \ &= \boxed{\binom{4}{1}}. \end{align*}

bingboyOct 12, 2023

#1

+789

-1

The sum of all the entries in the row of Pascal's triangle is 1+n+2n+⋯+2n+1=2n(n+1). Since the average of the entries is 2, we have [\frac{2n(n + 1)}{n + 1} = 2,] which simplifies to n=6.

bingboyOct 12, 2023

#1

+789

-1

There are five possible paths from A to I, if each step must be in a right-ward direction:

A→B→C→E→F→H→I

A→B→D→E→F→H→I

A→B→D→G→H→I

A→C→D→F→H→I

A→C→D→G→H→I

This is because there are two choices for the first step (to B or C), two choices for the second step (to D or E), and two choices for the third step (to F or G). The remaining steps are then forced.

bingboyOct 12, 2023

#1

+789

+1

Cost of 1 taco: x Cost of 1 burrito: y

Equation 1: 3x+2y=7.40 Equation 2: 4x+y= 6.45

To solve for x and y, we can use elimination.

Multiply the top equation by -1:

-3x-2y=-7.40

Add this equation to the bottom equation:

x-y=-1

Divide both sides by 1 to solve for x:

x=1

Substitute 1 for x in the top equation:

3(1)+2y=7.40

3+2y=7.40

2y=4.40

y=2.20

Therefore: Cost of 1 taco: 1 Cost of 1 burrito: 2.20

To calculate the cost of 3 tacos and 4 burritos:

(1)(3)+(2.2)(4)=11.80

Therefore, the cost of 3 tacos and 4 burritos is $11.80.

bingboyOct 12, 2023

#1

+789

-2

Let the height of triangle ABC be h.

[asy]

pair A, B, C, D, E;

A = (0,0);

B = (16,0);

C = (8,h);

D = (0,h-5);

E = (16,h-11);

draw(A--B--C--cycle);

draw(D--E);

draw(A--D,dashed);

draw(B--E,dashed);

label("A", A, SW);

label("B", B, SE);

label("C", C, N);

label("D", D, SW);

label("E", E, SE);

label("5", (A+D)/2, W);

label("11", (B+E)/2, E);

label("h", (C+D)/2, W);

label("h−5", (C+E)/2, E);

[/asy]

Since the shaded portions have the same area, we have the following equation:

(1/2) * 11 * h = (1/2) * (h - 5) * 16

Solving for h, we get:

h = 15

Therefore, the height of triangle ABC is 15 units.

bingboyOct 12, 2023

#1

+90

0

For the second part, it's 12!, which is 12*11*10*...*2*1, which is 479,001,600

totalkoolnezzOct 12, 2023

#1

+130511

+1

B - 50 - (B - 50)(1/3) = (3/7)B

B - 50 - (1/3)B+ 50/3 = (3/7)B

(2/3)B -100/3 = (3/7)B

(2/3 - 3/7)B = 100/3

(5/21)B = 100/3

B = (100/3) ( 21/5) = (100 / 5) ( 21 / 3) = 20 * 7 = 140 = total that he made

CPhillOct 11, 2023

#1

+130511

+1

D + J + K = 720 ⇒ J + K = 720 - D (1)

Derrick used .20D and had .80D left

John had J - 80 left

Kenneth had K - .40D left

.8D + J - 80 + K - .4D = 448 simplify

.4D + J + K = 448 + 80 ,...... sub in (1)

.4D + (720 -D) = 528

-.6D + = 528 - 720

-.6D = -192

D = -192 /( -.6D) = 320 = number Derrick had at first

CPhillOct 11, 2023

#1

+130511

+1

Call the total time = T

First room = (1/5)T

Second room = (T - 1/5 T) (1/3) = (4/5)T (1/3) = (4/15)T

Third room = 8

So

(1/5)T + (4/15)T + 8 = T

(7/15)T + 8 = T

8 = T - (7/15)T

8 = (8/15)T

T = 8 ( 15/8) = 15 hours

CPhillOct 11, 2023

#1

+130511

+1

Call the original number of sweets = S

Call the sweets he gave to his friends on Thursday = T

(S - T) = sweets left after Thursday

Sweets given on Thursday = (3/5) of what was left

T = (3/5)(S-T) simplify

(8/5)T = (3/5)S

8T = 3S

T = (3/8)S (1)

Sweets given on Friday = 130

Sweets given over 2 days = 4 more the (9/16) of what he started with

T + 130 = (9/16)S + 4

T + 126 = (9/16)S (2)

Sub (1) into (2)

(3/8)S + 126 = (9/16)S

126 = (9/16 - 3/8)S

126 = (9/16 - 6/16)S

126 = (3/16)S

S = 126 * 16/3 = 672 = number of sweets Edi had at first

Sweets given after 2 days = 672 - [ (9/16)(672) + 4 ] = (7/16)(672) - 4 = 290

Sweets left after 2 days = 672 - 290 = 382

CPhillOct 11, 2023

#1

+130511

+1

Rate * Time = Distance

Call the outward rate = S

Call the inward rate S+ 4

So

S( 2.5 hrs) = (S + 4) (2.25 hrs)

2.5 S = 2.25S + 9

2.5S - 2.25S = 9

.25S = 9

S = 9 / .25 = 36 km/h

S + 4 = 40km/h

The avg rate = 2 ( product of the rates) / ( sum of the rates) =

2 (36 * 40) / ( 36 + 40) =

2 (1440) / ( 76) ≈ 37.9 km/h

CPhillOct 11, 2023

#3

+130511

+1

y = x^2 / (x -1)

Rewrite as

y = x^2 ( x -1)^(-1) take the derivative, set to 0

y' = 2x (x -1)^(-1) - x^2 (x-1)^(-2) = 0

y' = (x-1)^(-2) [ 2x ( x -1) - x^2] = 0

y' = 2x^2 - 2x -x^2 = 0

y' = x^2 - 2x = 0

x ( x -2) = 0

x = 0 or x = 2

At x = 1.5, y = 4.5 at x =3, y = 4.5

So ....wnen x > 1, the min occurs at x = 2 { min = 4 }

CPhillOct 11, 2023

#1

+130511

+1

The x intercept occurs when y = 0

So

y = 8x + 14

0 = 8x + 14

-14 = 8x

-14/8 = x = -7/4 = the x intercept

CPhillOct 11, 2023

#1

+130511

+1

All the points are on the same line...So, the slope between any two points will be the same

So

[ 12 - -4] / [ -3 - -2 ] = 16 / -1 = -16

And

[ y - 12 ] / [ 4 - -3] = -16

[y -12 ] / [ 7] = -16

y -12 = -16 * 7

y - 12 = -112

y = -112 + 12 = -100

CPhillOct 11, 2023

#1

+130511

+1

Rational Roots Theorem

Possible factors of 28 = ±1, ±2 , ±4, ±7, ±14, ±28

Possible factors of 21 = ±1, ±3, ±7, ±21

Possible rational roots =

All possible factors of 28 / All possible factors of 21 =

±1, ±2, ±4, ±7, ±14, ±28, ±1/3 , ±2/3, ±4/3 , ± 4/3, ±7/3, ±14/3, ±28/3, ± 1/7, ±2/7,±4/7, ±1/21, ±2/21, ±4/21,

CPhillOct 11, 2023

#1

+789

-2

Let g(x)=f(x)−10. Then g(p)=g(q)=g(r)=g(s)=0, and so g is divisible by (x−p)(x−q)(x−r)(x−s). We can find this polynomial by taking the Chinese Remainder Theorem with modulus p−q on g(q), modulus p−r on g(r), and modulus p−s on g(s). This gives us $g(x)=10(x−p)(x−q)(x−r)(x−s)+100.$Hence, $f(x)=10(x−p)(x−q)(x−r)(x−s)+110.$By Vieta's formulas, the sum of the roots of x4−10x3+35x2−50x+24=0 is 10. Hence, p+q+r+s=10. By AM-GM, $(p+q+r+s)/4≥(pqrs)1/4=101/4=2,$so p+q+r+s≥8. Hence, $f(t)=10(t−p)(t−q)(t−r)(t−s)+110≥10(8)(t−8)+110=80t−660.

The minimum value of this expression is 5040, which occurs when t=11. Therefore, the answer is 5040.

bingboyOct 11, 2023

#2

+33661

+2

.As Melody often notes, it can be useful to plot these sort of probabilities against each other. Consider x_1 and x_2:

The probabilities of the pair can be thought of as uniformly randomly scattered points within a unit square.

The only regions of the square in which the lengths between the two values are greater than 3/4, are within the two red triangles indicated in the image above.

The probability that the lengths are greater than 3/4 is therefore given by p = Area of triangles/Area of square

The same probability holds for the lengths between x_3 and x_4, so, because the two probabilities are independent, the overall probability is given by p^2.

I'll leave you to do the number crunching.

AlanOct 11, 2023

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