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  s2=(a/2)2+v2=(4/2)2+32=13 s=(a/2)2+v2=(4/2)2+32=13 m≐3.6056 m  (v−h):r=v:a/2 v=h+r⋅ v:a/2 h=v−r⋅ 2v/a  V=πr2 h V=πr2(v−r⋅ 2v/a) V=πr2(v−r⋅ 2v/a) V=πr2(3−r⋅ 2⋅ 3/4) V′=3/2π(a−vr)r V′=0  3/2π⋅ (a−vr)r=0   3/2⋅ 3.1415926⋅ (4−3r)r=0 −14.1371667r2+18.85r=0 14.1371667r2−18.85r=0  a=14.1371667;b=−18.85;c=0 D=b2−4ac=18.852−4⋅14.1371667⋅0=355.305746317 D>0  r1,2=−b±D2a=18.85±355.3128.2743334 r1,2=0.66666667±0.666666666667 r1=1.33333333333 r2=0   Factored form of the equation:  r=14.1371667(r−1.33333333333)r=0r=r1=1.3333=43≐1.3333=1.333  m 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⋅ 2⋅ v/a=3−1.3333⋅ 2⋅ 3/4=1.0005=1.001  m h = v - r \cdot \ 2 \cdot \ v/a = 3 - 1.3333 \cdot \ 2 \cdot \ 3/4 = 1.0005 = 1.001 \ \text { m }

Try another example






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

Showing 0 comments:
1st comment
Be the first to comment!
avatar




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. Triangular pyramid
    TriangularPyramid What is the volume of a regular triangular pyramid with a side 3 cm long?
  2. Completing square
    eq2_5 Solve the quadratic equation: m2=4m+20 using completing the square method
  3. Quadratic equation
    Parabola_tangent Quadratic equation ? has roots x1 = -26 and x2 = -86. Calculate the coefficients b and c.
  4. Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  5. Square root 2
    parabola_2 If the square root of 3m2 +22 and -x = 0, and x=7, what is m?
  6. Equation 23
    reciprocal_1 Find value of unknown x in equation: x+3/x+1=5 (problem finding x)
  7. Calculate
    equilateral_triangle2 Calculate the length of a side of the equilateral triangle with an area of 50cm2.
  8. Quadratic equation
    kvadrat_2 Find the roots of the quadratic equation: 3x2-4x + (-4) = 0.
  9. Solve 3
    eq2_4 Solve quadratic equation: (6n+1) (4n-1) = 3n2
  10. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
  11. Evaluation of expressions
    eq222_10 If a2-3a+1=0, find (i)a2+1/a2 (ii) a3+1/a3
  12. Roots
    parabola Determine the quadratic equation absolute coefficient q, that the equation has a real double root and the root x calculate: ?
  13. Calculation
    pocty How much is sum of square root of six and the square root of 225?
  14. Catheti
    pyt_theorem The hypotenuse of a right triangle is 41 and the sum of legs is 49. Calculate the length of its legs.
  15. Solve equation
    hrusky_2 solve equation: ?
  16. Thunderstorm
    blesk The height of the pole before the storm is 10 m. After a storm when they come to check it they see that on the ground from the pole blows part of the column. Distance from the pole is 3 meters. At how high was the pole broken? (In fact, a rectangular tria
  17. Discriminant
    Quadratic_equation_discriminant Determine the discriminant of the equation: ?