# Secret treasure

Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.

Result

r =  1.333 m
h =  1.001 m

#### Solution:

$a = 4 \ m \ \\ v = 3 \ m \ \\ \ \\ s^2 = (a/2)^2 + v^2 = (4/2)^2 + 3^2 = 13 \ \\ s = \sqrt{ (a/2)^2 + v^2 } = \sqrt{ (4/2)^2 + 3^2 } = \sqrt{ 13 } \ m \doteq 3.6056 \ m \ \\ \ \\ (v-h):r = v:a/2 \ \\ v = h + r \cdot \ v:a/2 \ \\ h = v - r \cdot \ 2v/a \ \\ \ \\ V = \pi r^2 \ h \ \\ V = \pi r^2 (v-r \cdot \ 2v/a) \ \\ V = \pi r^2 (v-r \cdot \ 2v/a) \ \\ V = \pi r^2 (3-r \cdot \ 2 \cdot \ 3/4) \ \\ V' = 3/2 \pi (a-vr)r \ \\ V' = 0 \ \\ \ \\ 3/2 \pi \cdot \ (a-vr)r = 0 \ \\ \ \\ \ \\ 3/2 \cdot \ 3.1415926 \cdot \ (4-3r)r = 0 \ \\ -14.1371667r^2 +18.85r = 0 \ \\ 14.1371667r^2 -18.85r = 0 \ \\ \ \\ a = 14.1371667; b = -18.85; c = 0 \ \\ D = b^2 - 4ac = 18.85^2 - 4\cdot 14.1371667 \cdot 0 = 355.305746317 \ \\ D>0 \ \\ \ \\ r_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 18.85 \pm \sqrt{ 355.31 } }{ 28.2743334 } \ \\ r_{1,2} = 0.66666667 \pm 0.666666666667 \ \\ r_{1} = 1.33333333333 \ \\ r_{2} = 0 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ r=14.1371667 (r -1.33333333333) r = 0r = r_{ 1 } = 1.3333 = \dfrac{ 4 }{ 3 } \doteq 1.3333 = 1.333 \ \text { m }$

Checkout calculation with our calculator of quadratic equations.

$h = v - r \cdot \ 2 \cdot \ v/a = 3 - 1.3333 \cdot \ 2 \cdot \ 3/4 = 1.0005 = 1.001 \ \text { m }$

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

Be the first to comment!

#### Following knowledge from mathematics are needed to solve this word math problem:

Looking for help with calculating roots of a quadratic equation? 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. Pythagorean theorem is the base for the right triangle calculator. See also our trigonometric triangle calculator.

## Next similar math problems:

1. Garden
Area of a square garden is 6/4 of triangle garden with sides 56 m, 35 m, and 35 m. How many meters of fencing need to fence a square garden?
2. Coordinates of square vertices
I have coordinates of square vertices A / -3; 1/and B/1; 4 /. Find coordinates of vertices C and D, C 'and D'. Thanks Peter.
3. Uphill garden
I have a garden uphill, increasing from 0 to 4.5 m for a length of 25 m, how much is the climb in percent?
4. A concrete pedestal
A concrete pedestal has a shape of a right circular cone having a height of 2.5 feet. The diameter of the upper and lower bases are 3 feet and 5 feet, respectively. Determine the lateral surface area, total surface area, and the volume of the pedestal.
5. Trapezoid MO
The rectangular trapezoid ABCD with right angle at point B, |AC| = 12, |CD| = 8, diagonals are perpendicular to each other. Calculate the perimeter and area of ​​the trapezoid.
6. A rhombus
A rhombus has sides of length 10 cm, and the angle between two adjacent sides is 76 degrees. Find the length of the longer diagonal of the rhombus.
7. Isosceles triangle 10
In an isosceles triangle, the equal sides are 2/3 of the length of the base. Determine the measure of the base angles.
8. The hemisphere
The hemisphere container is filled with water. What is the radius of the container when 10 liters of water pour from it when tilted 30 degrees?
9. Cuboid face diagonals
The lengths of the cuboid edges are in the ratio 1: 2: 3. Will the lengths of its diagonals be the same ratio? The cuboid has dimensions of 5 cm, 10 cm, and 15 cm. Calculate the size of the wall diagonals of this cuboid.
10. Body diagonal
Calculate the volume of a cuboid whose body diagonal u is equal to 6.1 cm. Rectangular base has dimensions of 3.2 cm and 2.4 cm
11. Hole's angles
I am trying to find an angle. The top of the hole is .625” and the bottom of the hole is .532”. The hole depth is .250” what is the angle of the hole (and what is the formula)?
12. Medians in right triangle
It is given a right triangle, angle C is 90 degrees. I know it medians t1 = 8 cm and median t2 = 12 cm. .. How to calculate the length of the sides?
13. Faces diagonals
If the diagonals of a cuboid are x, y, and z (wall diagonals or three faces) respectively than find the volume of a cuboid. Solve for x=1.2, y=1.7, z=1.45
14. Wall and body diagonals
Calculate the lengths of the wall and body diagonals of the cuboid with edge dimensions of 0.5 m, 1 m, and 2 m
15. Two chords
Calculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.
16. Angles of elevation
From points A and B on level ground, the angles of elevation of the top of a building are 25° and 37° respectively. If |AB| = 57m, calculate, to the nearest meter, the distances of the top of the building from A and B if they are both on the same side of t
17. Space diagonal
The space diagonal of a cube is 129.91 mm. Find the lateral area, surface area and the volume of the cube.