A spherical tank is 3 m tall and therefore has a capacity of about 14000 L (NOTE that 3m is its diameter, and one cubic meter is 1000 L). You have been asked to put level lines on the tank showing the depths where the tank contains 1000, 5000, and 10000 L of liquid. (a) Use calculus to determine the height of each line (to the nearest cm). (b) Plot the circular cross-section of the tank and the level lines (NOTE the lines do not have to stay within the circle).

Answer:

(a) 1) When V = 1000 L, the height of the liquid in the tank is approximately 62 cm

2) When V = 5,000 L, the height of the liquid in the tank is approximately 106 cm

3) When V = 10,000 L, the height of the liquid in the tank is approximately 202 cm

b) The drawing of the spherical tank showing the height of the liquid level for each volume created with Microsoft Excel is attached

Step-by-step explanation:

(a) The height of the spherical tank = 3 m

The diameter of the spherical tank, d = 3 m

The capacity of the spherical tank = 14,000 L

The volume of a sphere, V = (4/3) × π × r³ =

Where, r = d/2

∴ r = 3 m/2 = 1.5 m

V = (4/3) × π × (1.5)³ ≈ 14.1371669412 m³

Using calculus, we have;

[tex]V = 2 \times \pi \times \int\limits^r_{0} {(r^2 - y^2)} \, dx = 2 \times \pi \times \left[r^2 \cdot y - \dfrac{y^3}{3} \right]^y_0[/tex]

For the two halves, we have;

[tex]V = \pi \times \left[r^2 \cdot y - \dfrac{y^3}{3} \right]^{\dfrac{y}{2}} _0 + \pi \times \left[r^2 \cdot y - \dfrac{y^3}{3} \right]^y_{\dfrac{y}{2}}[/tex]

[tex]V = 2 \times \pi \times \left[y^3- \dfrac{y^3}{3} \right]^y_0 = \dfrac{4 \cdot \pi \cdot y^3}{3}[/tex]

1) For V = 1000 L, we have;

[tex]1000 \ L = \dfrac{4 \cdot \pi \cdot y^3}{3}[/tex]

1000 L =

When V = 1000 L, we have;

[tex]y = \sqrt[3]{\dfrac{3 \times 1 \, m^3}{4 \times \pi } } \approx 0.62 \ m[/tex]

The liquid level, y ≈ 0.62 m = 62 cm

2) When V = 5,000 L, we have;

[tex]y = \sqrt[3]{\dfrac{3 \times 5 \, m^3}{4 \times \pi } } \approx 1.06 \ m[/tex]

The liquid level, y ≈ 1.06 m = 106 cm

3) When V = 10,000 L, we use the height of the unfilled volume at the top which is the volume of the sphere less the 10,000 L as follows;

For the volume of the sphere of 14,000 L, we have;

Volume at the top = 14,000 L - 10,000 L = 4,000 L = 4 m³

Therefore

[tex]y_{top} = \sqrt[3]{\dfrac{3 \times 4 \, m^3}{4 \times \pi } } \approx 0.985 \ m[/tex]

The height of the liquid in the tank, y = d - [tex]y_{top}[/tex]

∴ y = 3 m - 0.985 m = 2.015 m

The liquid level, y ≈ 2.015 m = 201.5 cm ≈ 202 cm

b) Please find attached the drawing of the spherical tank created with Microsoft Excel

a) (i) The level line for 1000 liters is set at a height of 0.486 meters.

(ii) The level line for 5000 liters is set at a height of 1.203 meters.

(iii) The level line for 10000 liters is set at a height of 1.926 meters.

b) The sphere is represented by the equation [tex]x^2+(y-1.5)^{2} = 2.25[/tex] and the level lines are represented by the linear functions [tex]y = 0.486[/tex], [tex]y = 1.203[/tex] and [tex]y = 1.926[/tex]. Please see the second image attached.

a) First, we derive an expression for the volume of a spherical section ([tex]V[/tex]), in cubic meters, in terms of tank radius ([tex]R[/tex]), in meters, and liquid height ([tex]h[/tex]), in meters. The volume can be found by using the slicing method integral equation:

[tex]V = \int\limits^{H}_{0} {A(h)} \, dh[/tex] (1)

Where:

If we know that [tex]A = \pi\cdot [R^{2}-(R-h)^{2}][/tex], then the volume formula is:

[tex]V = \pi\int\limits^H_0 {h\cdot (2\cdot R-h)} \, dh[/tex]

[tex]V = 2\pi\cdot R\int\limits^{H}_{0} {h} \, dh - \pi \int\limits^{H}_{0} {h^{2}} \, dh[/tex]

[tex]V = \pi\cdot R\cdot H^{2}-\frac{\pi\cdot H^{3}}{3}[/tex]

Then we obtain the following third order polynomial:

[tex]\frac{\pi\cdot H^{3}}{3}-\pi\cdot R\cdot H^{2}+V = 0[/tex] (2)

Now we proceed to solve numerically for each case:

[tex]H = 0.486\,m[/tex]

The level line for 1000 liters is set at a height of 0.486 meters. [tex]\blacksquare[/tex]

[tex]H = 1.203\,m[/tex]

The level line for 5000 liters is set at a height of 1.203 meters. [tex]\blacksquare[/tex]

[tex]H = 1.926\,m[/tex]

The level line for 10000 liters is set at a height of 1.926 meters. [tex]\blacksquare[/tex]

b) The sphere is represented by the equation [tex]x^2+(y-1.5)^{2} = 2.25[/tex]level lines are represented by the linear functions [tex]y = 0.486[/tex], [tex]y = 1.203[/tex] and [tex]y = 1.926[/tex]. By a graphing tool we have the following result. [tex]\blacksquare[/tex]

To learn more on volumes, we kindly invite to check this verified question: https://brainly.com/question/1578538

if the cube was Rolled 78 x and the nine appears 26 times what was the experimental probability of rolling a 9​

expand the following expression by using the distributive method 7(10-3x))

David had a coupon for the grocery store. He graphed on the x-axis what his total bill would be if he didn’t use the coupon and graphed on the y-axis what his total bill would be if he used the coupon.

Grocery Bill
A graph has bill without coupon on the x-axis and bill with coupon on the y-axis. A line goes through points (2, 0) and (3, 1).

Based on the information in the graph, if David’s bill came to $66 after the coupon was applied, how much would his bill have been if he hadn’t used the coupon?
$64
$65
$67
$68
I will give brainliest

Can help find the answer , what the increasing interval and decreasing interval. In interval notations?

3
Choose all the expressions that are equal to
A.
8
15
5
11
B.
7
12
+
6
C. 5
6
-
1
12
D.
5. 1+
E.
7
20
3
5

WHAT I CAN DO
Activity 4: Solve the following problem.
1. Ellen has 2 chicken eggs and 3 quail eggs in a tray. What is the ratio of quail eggs
to chicken eggs?
2. A basket contains 75 mangoes. What is the ratio of the basket to the total
number of mangoes?
3. Kris bought 2 pairs of shoes and 5 pairs of socks. What is the ratio of shoes to
socks?
4. Rina planted 15 tomato seedlings and 20 pechay seedlings. What is the ratio of
pechay seedlings to tomato seedlings?
5. What is the ratio of vowel letters to consonant letters in your full name?​

4/5 ÷ 2/5 = ?????????​

CAN SOME PLEASE HELP ITS AN I EMERGENCY

how many eighth notes are in a sixteenth note

What percent of 50 is 10?

A car manufacturer announced that next year the price of a certain model car would increase by 3.5%. This year the price is $15,649 . The increase in the price of the car is $? The new price of the car is $?

Given that 6x – 7y = 22
Find y when x = -1
y =

Which polynomial represents the quotient for the division problem shown?
-1
3
-4
-3
-7
2
7
9
-1
-9
-10
3
3
O A. 3.x2 – 7x+9
O B. - 3x² + 7x – 9
O C. 3.rº – 7x2 + 9x - 10
O D. 3.x® – 4x + 2x - 1

3. Find the value of x.
Need help will mark Brainliest

4k+1/2-2

Can Anyone help me

how do you describe the graph of |x| < 12 brainly

Suppose you had $6.00 to buy bananas and apples. Bananas cost $0.49 per pound and apples cost $0.34 per pound. Write a linear equation that represents the different amounts of fruit you could buy.​

Solve the equation: p2 - 4p + 4 = 0.​

24 - 3x Evaluate each expression if x = 5

Stores often mark up items each year to cover inflation.
Last year, a CD at Juan's Tunes cost $12.95. This year,
the cost was marked up 8%. The store also gave anyone
who bought four CDs this
year a 10% discount. Explain
the steps to find the final cost for the four CDs this year.

sin(30)=1
the solution set is{}?

4x-5=13x-2

I’m stuck on this problem

Please help ASAP! please and thank you!! Have a great and blessed day!

It is possible, through careful consideration of economic principles and interactions, to reach a state of equilibrium in practice.


Please select the best answer from the choices provided

T
F

PLEASE HELP ME PLSSSSS!!! The question is on the photo!!!

2. Find the product using any property 4x7649x25.

​

Find the value of the discriminant of the following equation 10x² + x = 1.​

Apples and tangerines were sold in the ratio of 6 to 4.
40
tangerines were sold. How many fruits were sold?
​

Latoya's softball team won 75%, or 18, of its game. Define a variable. Write and solve an equations to determine the number of games the team played.
Will give brainliest

give the name monomial, binomial, trinomial and the degree of the polynomial
x.