Nick has 3 shirts: a white one, a black one, and a blue one. He also has two pairs of pants, one blue and one tan. What is the p
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Answer:
Step-by-step explanation:
Answer: = 1087.5
Step-by-step explanation: To evaluate the triple integral, first an equation of a plane is needed, since the tetrahedon is a geometric form that occupies a 3 dimensional plane. The region of the integral is in the attachment.
An equation of a plane is found with a point and a normal vector. <u>Normal</u> <u>vector</u> is a perpendicular vector on the plane.
Given the points, determine the vectors:
P = (5,0,0); Q = (0,9,0); R = (0,0,4)
vector PQ = (5,0,0) - (0,9,0) = (5,-9,0)
vector QR = (0,9,0) - (0,0,4) = (0,9,-4)
Knowing that cross product of two vectors will be perpendicular to these vectors, you can use the cross product as normal vector:
n = PQ × QR =
n = 36i + 0j + 45k - (0k + 0i - 20j)
n = 36i + 20j + 45k
Equation of a plane is generally given by:
Then, replacing with point P and normal vector n:
The equation is: 36x + 20y + 45z - 180 = 0
Second, in evaluating the triple integral, set limits:
In terms of z:
When z = 0:
When z=0 and y=0:
x = 5
Then, triple integral is:
Calculating:
= 1087.5
<u>The volume of the tetrahedon is 1087.5 cubic units.</u>
Answer:
Step-by-step explanation:
- The equation of the line in point-slope form is:
Where m is the slope and is a point of the line.
- The equation of the line is written is slope-intercept form, where the slope is
.
- By definition, when two lines are perpendicular to each other, the slopes are the negative reciprocals. Therefore, the slope of the perpendicular line must be -3.
- Substitute the point P and the slope -3 into the equation in point-slope form, therefore, you obtain: