The base of a triangular pyramid is a triangle with corners at #(6 ,7 )#, #(3 ,1 )#, and #(5 ,9 )#. If the pyramid has a height of #8 #, what is the pyramid's volume?

Answer 1

#color(green)(V = 16.5781)# cubic units

Triangle Area #A = (1/2) * a * h #

#BC = a = sqrt((5-3)^2 + (9-1)^2) = 8.2462#

Slope of BC #m_a = (9-1) / (5-3) = 4#

Equation of BC

#(y - 1) / (9 - 1) = (x - 3) / (5 - 3)#

#2y - 2 = 8x - 24#

#y - 4x = -11# Eqn (1)

Slope of altitude through A is #m__h = -1/4#

Equation of altitude h is

#y - 7 = -(1/4) (x - 6)#

#4y + x = 34# Eqn 2

Solving equations (1) & (2), we get the coordinates of D, the base of altitude h

#D (78/11, 128/17)

Height of altitude #AD = h = sqrt((6-(78/17))^2 + (7 - (128/17))^2) = 1.5078#

Area of Triangle #color(green)A = (1/2) b h = (1/2) * 8.2462 * 1.5078 = color(green)(6.2168)#

Volume of Pyramid V = (1/3) * Base Area * Height

#color(green)(V = (1/3) 6.2168 * 8 = 16.5781)#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the volume of the triangular pyramid, you can use the formula ( V = \frac{1}{3} \times \text{base area} \times \text{height} ).

First, find the area of the base triangle using the coordinates given for its vertices. Then, calculate the distance between the apex of the pyramid and the base plane to determine the height. Finally, substitute the values into the formula to find the volume.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7