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2 years ago

If $x$, $y$, and $z$ are positive integers such that $6xyz+30xy+21xz+2yz+105x+10y+7z=812$, find $x+y+z$.

Mathematics

1 answer:

harina [27]2 years ago

8 0

Answer:

Step-by-step explanation:

When we simplify we get $z(6xy+21x+2y+7)+30xy+105x+10y=812.$

Then we continue to factor to get: $(z+5)(6xy+21x+2y+7)&=847.$

We then see that we can factor $(6xy+21x+2y+7)$ into $(z+5)(3x+1)(2y+7)&=847.$

we then do the prime factorization of 847, which i think is, $7*11*11$ . we have to find the numbers that multiply to 847 and then plug them into z+5, 3x=1,2y+7.

It has to be a positive, non-negative integer, right?

We also see that 3x+1=11 so we see that x=10/3 (which wont work).

So 3x+1=7, so x=2.

So 11 has to be in another term. It has to be in 2y+7=11 so y=2

for the last term we get z+5=11 so z=6

2+2+6=10

Hope this helps and if you want please consider giving me brainliest. :)

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Match each three-dimensional figure to its volume based on the given dimensions. (Assume π = 3.14.)

LekaFEV [45]

Answer:

The volume of the cylinder is 150.72 cm³ ⇒ last answer

The volume of the cone is 314 cm³ ⇒ 1st answer

The volume of the pyramid is 160 cm³ ⇒ 2nd answer

The volume of the pyramid is 48 cm³ ⇒ 3rd answer

Step-by-step explanation:

* Lets revise the volumes of some shapes

- The volume of the cylinder of radius r and height h is:

V = π r² h

- The volume of the cone of radius r and height h is:

V = 1/3 π r² h

- The volume of the pyramid is:

V = 1/3 × its base area × its height

* Lets solve the problem

# A cylinder with radius 4 cm and height 3 cm

∵ V = π r² h

∵ π = 3.14

∵ r = 4 cm , h = 3 cm

∴ v = 3.14 (4)² (3) = 150.72 cm³

* The volume of the cylinder is 150.72 cm³

# A cone with radius 5 cm and height 12 cm

∵ V = 1/3 π r² h

∵ π = 3.14

∵ r = 5 cm , h = 12 cm

∴ V = 1/3 (3.14) (5)² (12) = 314 cm³

* The volume of the cone is 314 cm³

# A pyramid with base area 16 cm² and height 30 cm

∵ V = 1/3 × its base area × its height

∵ The area of the base is 16 cm²

∵ The height = 30 cm

∴ V = 1/3 (16) (30) = 160 cm³

* The volume of the pyramid is 160 cm³

# A pyramid with square base of length 3 cm and height 16 cm

∵ V = 1/3 × its base area × its height

∵ The area of the square = s²

∵ The area of the base = 3² = 9 cm²

∵ The height = 16 cm

∴ V = 1/3 (9) (16) = 48 cm³

* The volume of the pyramid is 48 cm³

3 0

1 year ago

Read 2 more answers

Pudge has three times as many apples as Ace, and Ace has twice as many as Christi. How many apples does each have if pudge has 1

mote1985 [20]

Answer:

<h3>Pudge has 12 morethan apples that is 24 apples and both Ace and Christi has not more than 12 apples together they has atmost 12 apples.The apples does each having is</h3><h3>P = Pudge's Apples =24</h3><h3>A = Ace's Apples =8</h3><h3>and C = Christi's Apples=4.</h3>

Step-by-step explanation:

Let P = Pudge's Apples

Let A = Ace's Apples

Let C = Christi's Apples

<h3>To find how many apples does each have if Pudge has 12 more than both Ace and Christi together:</h3>

Given that P = 3A , A = 2C and P = A + C + 12

Substitute the value for P in P = A + C + 12 we get

3A = A + C + 12

3A-A=C+12

2A=C+12

From A = 2C we have that $C=\frac{A}{2}$

Substitute the value C:

$2A=\frac{A}{2}+12$

$2A-\frac{A}{2}=12$

$\frac{4A-A}{2}=12$

$\frac{3A}{2}=12$

$3A=12\times 2$

$A=\frac{24}{3}$

Substituting the value of A in P=3A we get

P = 3(8)

Substituting the values of P and A in P = A + C + 12

$24=8+C+12$

$24=C+20$

$24-20=C$

4=C

Rewritting we get

<h3>∴ C=4 </h3><h3>Hence Pudge has 12 morethan apples that is 24 apples </h3><h3>and both Ace and Christi has not more than 12 apples together they has atmost 12 apples</h3>

7 0

1 year ago

A pizza parlor offers 4 different pizza toppings. How many different kinds of 2-topping pizzas are available?

Galina-37 [17]

Order does not matter so use "n choose k" formula is used to find number of unique combinations.

c=n!/(k!(n-k)!) where n is total possible choices and k is number of selections.

c=4!/(2!(4-2)!)

c=4!/(2!2!)

c=24/(2*2)

c=24/4

c=6

So there are 6 different two topping options when there are four different toppings to choose from.

6 0

2 years ago

Read 2 more answers

During an experiment, Juan rolled a six-sided number cube 18 times. The number two occurred four times. Juan claimed the experim

natita [175]

Answer:

$\frac{2}{9}$