Conical bottle

When a conical bottle rests on its flat base, the water in the bottle is 8 cm from it vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle?

Result

h =  10.22 cm

Solution:

$h_{1}=8 \doteq 10.2195 \ \text{cm} \ \\ h_{2}=2 \doteq -8.2195 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \pi r^2 \ h \ \\ V_{3}=\dfrac{ 1 }{ 3 } \pi r_{1}^2 \ h_{1} \ \\ \ \\ V_{1}=V-V_{3} \ \\ V_{1}=\dfrac{ 1 }{ 3 } \pi r^2 \ h-\dfrac{ 1 }{ 3 } \pi r_{1}^2 \ h_{1} \ \\ V_{1}=\dfrac{ 1 }{ 3 } \pi (r^2 \ h - r_{1}^2 \ h_{1}) \ \\ \ \\ r_{1}0 \ \\ \ \\ h_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 2 \pm \sqrt{ 340 } }{ 2 }=\dfrac{ 2 \pm 2 \sqrt{ 85 } }{ 2 } \ \\ h_{1,2}=1 \pm 9.2195444572929 \ \\ h_{1}=10.219544457293 \ \\ h_{2}=-8.2195444572929 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (h -10.219544457293) (h +8.2195444572929)=0 \ \\ \ \\ h>0 \ \\ \ \\ h=h_{1}=10.2195 \doteq 10.2195 \doteq 10.22 \ \text{cm}$

Checkout calculation with our calculator of quadratic equations.

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?
Check out our ratio calculator.
Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?
Tip: Our volume units converter will help you with the conversion of volume units.

Next similar math problems:

1. Line
Straight line passing through points A [-3; 22] and B [33; -2]. Determine the total number of points of the line which both coordinates are positive integers.
2. Parametric form
Calculate the distance of point A [2,1] from the line p: X = -1 + 3 t Y = 5-4 t Line p has a parametric form of the line equation. ..
3. Find the 5
Find the equation with center at (1,20) which touches the line 8x+5y-19=0
4. Right triangle from axes
A line segment has its ends on the coordinate axes and forms with them a triangle of area equal to 36 square units. The segment passes through the point ( 5,2). What is the slope of the line segment?
5. Coordinates of square vertices
The ABCD square has the center S [−3, −2] and the vertex A [1, −3]. Find the coordinates of the other vertices of the square.
6. Right angled triangle 2
LMN is a right angled triangle with vertices at L(1,3), M(3,5) and N(6,n). Given angle LMN is 90° find n
7. Find the 10
Find the value of t if 2tx+5y-6=0 and 5x-4y+8=0 are perpendicular, parallel, what angle does each of the lines make with the x-axis, find the angle between the lines?
8. Three points 2
The three points A(3, 8), B(6, 2) and C(10, 2). The point D is such that the line DA is perpendicular to AB and DC is parallel to AB. Calculate the coordinates of D.
9. Vector equation
Let’s v = (1, 2, 1), u = (0, -1, 3) and w = (1, 0, 7) . Solve the vector equation c1 v + c2 u + c3 w = 0 for variables c1 c2, c3 and decide weather v, u and w are linear dependent or independent
10. Angle of the body diagonals
Using vector dot product calculate the angle of the body diagonals of the cube.
11. Coordinates of a centroind
Let’s A = [3, 2, 0], B = [1, -2, 4] and C = [1, 1, 1] be 3 points in space. Calculate the coordinates of the centroid of △ABC (the intersection of the medians).
12. Set of coordinates
Consider the following ordered pairs that represent a relation. {(–4, –7), (0, 6), (5, –3), (5, 2)} What can be concluded of the domain and range for this relation?
13. Two people
Two straight lines cross at right angles. Two people start simultaneously at the point of intersection. John walking at the rate of 4 kph in one road, Jenelyn walking at the rate of 8 kph on the other road. How long will it take for them to be 20√5 km apa
14. Points collinear
Show that the point A(-1,3), B(3,2), C(11,0) are col-linear.
15. Cuboids
Two separate cuboids with different orientation in space. Determine the angle between them, knowing the direction cosine matrix for each separate cuboid. u1=(0.62955056, 0.094432584, 0.77119944) u2=(0.14484653, 0.9208101, 0.36211633)
16. Angle between vectors
Find the angle between the given vectors to the nearest tenth of a degree. u = (-22, 11) and v = (16, 20)
17. Slope form
Find the equation of a line given the point X(8, 1) and slope -2.8. Arrange your answer in the form y = ax + b, where a, b are the constants.