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.

Correct result:

V =  540 cm3

Solution:

S=468 cm2 a:b=3:4 h=b2  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)(4x2)  34x2+(3x+4x+5x)(4x2)=468  3 4 x2+(3x+4x+5x)(4x2)=468 60x224x468=0  a=60;b=24;c=468 D=b24ac=242460(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(x3)(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=b2=122=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*4*x^2 + (3x+4x+5x)(4x-2)=468 \ \\ \ \\ 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.




We would be very happy if you find an error in the example, spelling mistakes, or inaccuracies, and please send it to us. We thank you!






Showing 0 comments:
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.

 
We encourage you to watch this tutorial video on this math problem: video1   video2   video3

Next similar math problems:

  • Triangular prism
    hranol_3sides The base perpendicular triangular prism is a right triangle whose hypotenuse measures 5 cm and one cathetus 2 cm. Height of the prism is equal to 7/9 of the perimeter of the base. Calculate the surface area of prism.
  • 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.
  • 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.
  • Triangular prism - regular
    prism3s The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.
  • Triangular prism
    prism3 Calculate the surface of a triangular prism with the base of an equilateral triangle with a side length of 7.5 cm and a corresponding height of 6.5 cm. Prism height is 15cm.
  • Triangular pyramid
    3sidesPyramid 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)?
  • Quadrilateral pyramid
    jehlan_2 A regular quadrilateral pyramid has a volume of 24 dm3 and a base edge a = 4 dm. Calculate: a/height of the pyramid b/sidewall height c/surface of the pyramid
  • 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.
  • Embankment
    nasyp The railway embankment 300 m long has a cross section of an isosceles trapezoid with bases of 14 m and 8 m. The trapezoidal arms are 5 m long. Calculate how much m3 of soil is in the embankment?
  • 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.
  • 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).
  • 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
  • Tetrahedron
    tetrahedron (1) Calculate height and volume of a regular tetrahedron whose edge has a length 4 cm.
  • The right triangle
    rt_tr540 The right triangle ABC has a leg a = 36 cm and an area S = 540 cm2. Calculate the length of the leg b and the median t2 to side b.
  • 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.
  • 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.
  • 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?