Base of prism

The base of the perpendicular prism is a rectangular triangle whose legs length are at a 3: 4 ratio. The height of the prism is 2cm smaller than the larger base leg. Determine the volume of the prism if its surface is 468 cm2.

Result

V =  540 cm3

Solution:

S=468 cm2 a:b=3:4 h=b−2  a2+b2=c2 a=3x b=4x c=5x 32+42=52   S=ab+(a+b+c)h S=3⋅ 4⋅ x2+(3x+4x+5x)(4x−2)   3⋅ 4⋅ x2+(3x+4x+5x)(4x−2)=468 60x2−24x−468=0  a=60;b=−24;c=−468 D=b2−4ac=242−4⋅60⋅(−468)=112896 D>0  x1,2=−b±D2a=24±112896120 x1,2=24±336120 x1,2=0.2±2.8 x1=3 x2=−2.6   Factored form of the equation:  60(x−3)(x+2.6)=0  x>0 x=x1=3 cm  a=3⋅ x=3⋅ 3=9 cm b=4⋅ x=4⋅ 3=12 cm c=5⋅ x=5⋅ 3=15 cm h=b−2=12−2=10 cm  S2=a⋅ b+(a+b+c)⋅ h=9⋅ 12+(9+12+15)⋅ 10=468 cm2 S2=S  V=a⋅ b2⋅ h=9⋅ 122⋅ 10=540 cm3S=468 \ \text{cm}^2 \ \\ a:b=3:4 \ \\ h=b-2 \ \\ \ \\ a^2 + b^2=c^2 \ \\ a=3x \ \\ b=4x \ \\ c=5x \ \\ 3^2+4^2=5^2 \ \\ \ \\ \ \\ S=ab + (a+b+c)h \ \\ S=3 \cdot \ 4 \cdot \ x^2 + (3x+4x+5x)(4x-2) \ \\ \ \\ \ \\ 3 \cdot \ 4 \cdot \ x^2 + (3x+4x+5x)(4x-2)=468 \ \\ 60x^2 -24x -468=0 \ \\ \ \\ a=60; b=-24; c=-468 \ \\ D=b^2 - 4ac=24^2 - 4\cdot 60 \cdot (-468)=112896 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 24 \pm \sqrt{ 112896 } }{ 120 } \ \\ x_{1,2}=\dfrac{ 24 \pm 336 }{ 120 } \ \\ x_{1,2}=0.2 \pm 2.8 \ \\ x_{1}=3 \ \\ x_{2}=-2.6 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 60 (x -3) (x +2.6)=0 \ \\ \ \\ x>0 \ \\ x=x_{1}=3 \ \text{cm} \ \\ \ \\ a=3 \cdot \ x=3 \cdot \ 3=9 \ \text{cm} \ \\ b=4 \cdot \ x=4 \cdot \ 3=12 \ \text{cm} \ \\ c=5 \cdot \ x=5 \cdot \ 3=15 \ \text{cm} \ \\ h=b-2=12-2=10 \ \text{cm} \ \\ \ \\ S_{2}=a \cdot \ b + (a+b+c) \cdot \ h=9 \cdot \ 12 + (9+12+15) \cdot \ 10=468 \ \text{cm}^2 \ \\ S_{2}=S \ \\ \ \\ V=\dfrac{ a \cdot \ b }{ 2 } \cdot \ h=\dfrac{ 9 \cdot \ 12 }{ 2 } \cdot \ 10=540 \ \text{cm}^3

Checkout calculation with our calculator of quadratic equations.


Try another example


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...):

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




Tips to related online calculators
Pythagorean theorem is the base for the right triangle calculator.
Tip: Our volume units converter will help you with the conversion of volume units.
See also our trigonometric triangle calculator.

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

Next similar math problems:

  1. 3s prism
    Prism It is given a regular perpendicular triangular prism with a height 19.0 cm and a base edge length 7.1 cm. Calculate the volume of the prism.
  2. Triangular prism
    hranol_3 Base of perpendicular triangular prism is a right triangle with leg length 5 cm. Content area of the largest side wall of its surface is 130 cm² and the height of the body is 10 cm. Calculate its volume.
  3. Hexagon
    hexa_prism Calculate the surface of a regular hexagonal prism whose base edge a = 12cm and side edge b = 3 dm.
  4. Quadrangular prism
    hranol_4 The regular quadrangular prism has a base edge a = 7.1 cm and side edge = 18.2 cm long. Calculate its volume and surface area.
  5. Triangular pyramid
    ihlan_3b It is given perpendicular regular triangular pyramid: base side a = 5 cm, height v = 8 cm, volume V = 28.8 cm3. What is it content (surface area)?
  6. Tetrahedral pyramid
    jehlan A regular tetrahedral pyramid is given. Base edge length a = 6.5 cm, side edge s = 7.5 cm. Calculate the volume and the area of its face (side area).
  7. Quadrangular pyramid
    jehlan_4b_obdelnik_1 Given is a regular quadrangular pyramid with a square base. The body height is 30 cm and volume V = 1000 cm³. Calculate its side a and its surface area.
  8. Tetrahedral pyramid
    jehlan_4b_obdelnik_3 Calculate the surface S and the volume V of a regular tetrahedral pyramid with the base side a = 5 m and a body height of 14 m.
  9. Triangular prism
    hranol_5 The perpendicular triangular prism is a right triangle with a 5 cm leg. The content of the largest wall of the prism is 130 cm2 and the body height is 10 cm. Calculate the body volume.
  10. Triangular prism,
    prism3s The regular triangular prism, whose edges are identical, has a surface of 2514 cm ^ 2 (square). Find the volume of this body in cm3 (l).
  11. A concrete pedestal
    frustum-of-a-right-circular-cone 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.
  12. Free space in the garden
    euklid The grandfather's free space in the garden was in the shape of a rectangular triangle with 5 meters and 12 meters in length. He decided to divide it into two parts and the height of the hypotenuse. For the smaller part creates a rock garden, for the large
  13. Cone container
    kuzel_1 Rotary cone-shaped container has a volume 1000 cubic cm and a height 12 cm. Calculate how much metal we need for making this package.
  14. Wall height
    jehlan_2 Calculate the surface and volume of a regular quadrangular pyramid if side a = 6 cm and wall height v = 0.8dm.
  15. Tetrahedron
    tetrahedron (1) Calculate height and volume of a regular tetrahedron whose edge has a length 18 cm.
  16. 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?
  17. Holidays - on pool
    pool_4 Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?