A translation is a transformation where all points of the figure are moved the same distance in the same direction. The distance and direction are indicated by a ray sometimes called the translation vector. A vector is a quantity that has both length and direction, and can be thought of as a line segment with a starting point and an endpoint. A translation is an isometry, so the image of a translated figure is congruent to the preimage.
Often, the rule for the translation will be given as (x, y) → (x + a, y + b). This is a shift a units in the x-direction (horizontally) and b units in the y-direction (vertically). For instance, in the coordinate plane below the translation (x, y) → (x + 4, y – 2) shifts each point 4 units to the right and 2 units down.
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Find the volume of the tetrahedron having the given vertices. (3, −1, 1), (5, −2, 4), (1, 1, 1), (0, 0, 1)
Dafna11 [192]
Given four vertices of a tetrahedron, we need to find its volume.
A(3,-1,1), B(5,-2,4), C(1,1,1), D(0,0,1)
The simplest method is to use vectors.
Here, we calculate the volume of a parallelepiped defined by vectors AB, AC, AD. The volume of a tetrahedron is one-sixth of the volume of the parallelepiped, V, given by AB.(ACxAD).
Here are the steps:
1. Choose a vertex, say A. Calculate vectors AB, AC and AD
AB=(5-2,4)-(3,-1,1) = <2, -1, 3>
AC=(1,1,1)-(3,-1,1) = <-2, 2, 0>
AD=(0,0,1)-(3,-1,1) = <-3, 1, 0>
2. Calculate vector cross product ACxAD = <-2,2,0>x<-3,1,0>
i j k
-2 -2 0
-3, 1, 0
=<0-0, 0-0, -2-(-6)>
= <0, 0, 4>
3. Calculate volume of parallelepiped, equal to the dot product AB.(ACxAD), ignore sign, which is arbitrary.
V = |<2, -1, 3>.<0,0,4>| = |12| = 12
4. Calculate volume of tetrahedron, which is one-sixth of volume of parallelepiped
V(tetrahedron) = V/6= 12/6 = 2
Let me just.. take those.. thanks
You would have to replace f(x) with zero and use quadratic formula
f(x)=x^2-5x+5
0=x^2-5x+5
=(-b+-√b^2-4ac)/2a
=(5+-√25-4(1)(5))/2
=(5+-√5)/2
x=(5+√5)/2 or x=(5-√5)/2
Both x values are completely applicable in the equation.
Hope I helped :)
Answer: 12 dozens of cupcakes will be made after three hours.
Step-by-step explanation: If you got to the three under hours and move up until the arrow intersects you will get 12 dozens
