The volume of a cylinder can be determined using the formula V=πr2h, where r and h represent the radius and height of the cylinder, respectively. A volume of paint expressed as (8x3 + 31x2 + 32x)π and a volume of paint expressed as (10x3 + 17x2)π are poured into a paint can in the shape of a cylinder. Determine possible expressions for the radius of the can and the depth of the paint in the can.

Answer:

Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.

Step-by-step explanation:

Let be the initial volumes of the initial cans represented by these expressions:

[tex]V_{1} = (8\cdot x^{3}+31\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (1)

[tex]V_{2} = (10\cdot x^{3}+17\cdot x^{2})\cdot \pi[/tex] (2)

The resulting volume of the paint can is the sum of the two functions:

[tex]V_{3} = (18\cdot x^{3}+48\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (3)

Then, we proceed to factor the polynomial:

[tex]V_{3} = 2\cdot (9\cdot x^{2}+24\cdot x +16)\cdot x \cdot \pi[/tex]

[tex]V_{3} = \pi\cdot (9\cdot x^{2}+24\cdot x + 16)\cdot (2\cdot x)[/tex] (3b)

By direct comparison with the volume formula for the cylinder we have the following expressions:

[tex]r^{2} = 9\cdot x^{2}+24\cdot x + 16[/tex]

[tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex]

[tex]h = 2\cdot x[/tex]

Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.

ONLY ANSWER IF U KNOW!
Find the limit of the function algebraically.

Mario is buying a number of hamburgers from the local store that cost $2.90 each. He is also buying one packet of hamburger rolls at a cost of $4.75. He has $39.55 to spend at the store. Write and solve an inequality that shows how many hamburgers, h, Mario can afford to buy.
Write the inequality.

solve the inequality

Find the value of a and b when x = 10
5x2
2
2x²(x - 5)
10x

___ divided by 3x3 =12

im need help please!!!

Can someone help me with this problem

4 people can dig a trench in 3 hours.
How long would it take 9 people?
Give your answer in minutes.

does anyone here know how to write negative 1 on there

PLEASE HELP WITH MY HOMEWORK SCREEN SHOT ATTACTCHED

What is the area of this polygon?

simplify leaving your answer in the standard form
[tex] \frac{0.0225 \times 0.0256}{0.0015 \times 0.48} [/tex]
​

Please Help — Neil is going to a bookstore 45 miles away. The bridge was closed on the way back, so
he had to take an alternate route and had to drive 15 mph slower, which make the trip
back take 7 minutes longer. How fast was he going on the way to the bookstore??

What is the theoretical probability of rolling anything other than a two on a fair six sided die?

PLS HELP MEEEEEEEE I WILL GIVE BRAINLIEST :D

someone help please!! oh and pls don’t put any links

Joe has been riding his bicycle at a speed of 12 mph and has gone 48 miles. How long has he been riding?

Joe has 3 1/4 gallons of punch. He adds 1 1/2 quarts of juice to the punch . He drinks 1/4 quarts of punch . Which expression can be used to fine the number of quarts of punch joe has left

can someone please help me with this question?

what’s the answer ??

Always in the game and never played by the rules (rules)
Tried to let me leave (leave), fell down to my knees (knees)
Picked myself up and turned my back to the breeze, oh-oh
Spent a milli' on a milli', mama look at me (oh)
With all these diamond chokers, man, it's gettin' hard to breathe

A glass bead has the shape of a rectangular prism with a smaller rectangular prism removed. What is the volume of the glass that forms the​ bead?

Thanks in advance!

find the angles marked with a letter ​

The amount of carbon-14 in an object is given by y = ae– 0.00012t where a is the initial amount of carbon and t is the age in years. A fossil bone contains 25% of its original carbon-14. What is the approximate age of the bone?

what is the measure of ∠x?

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

Lin understands that expanding is using the distributive property, but she doesn’t understand what factoring is or why it works. How can Kiran explain factoring to Lin?

Ms. Padilla is ordering art supplies for 90 students. She
orders 8 jars of paint for every 5 students and 10 sketch
pads for every 3 students.
Determine whether each statement about Ms. Padilla's order is true or false.
Select True or False for each statement.
Ms. Padilla orders 40 sketch pads.
O True
O False
оо
Ms. Padilla orders 144 jars of paint.
O True
O False
The ratio of sketch pads to jars of paint in Ms. Padilla's order can be written as
25 : 12
O True
O False
The ratio of jars of paint to sketch pads in Ms. Padilla's order can be written as
4:5.
o
O True
O False

what is is 7 *10+ 7 i need to now please help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1

8/6 + (3/8 + x)(2) =

Write a [tex]y=\frac{4}{5}x-2[/tex] in standard form using integers.