carly bought a new house for $125,000. The value of the house appreciates approximately 3.5% each year. What will be the value o
1 answer:
You might be interested in
Answer:
Option D is correct.
VS = 20 units.
Step-by-step explanation:
The complete question is contained in the attached image to this answer.
From the image attached and the additional information about the triangles, it is evident that that triangle VSW is similar to triangle QSR.
Hence, we can write the ratio as
(VS/QS) = (VW/QR)
VS = 2a
QS = 2 × VS = 2 × 2a = 4a
VW = 2a - 2
QR = 3a + 6
(2a)/(4a) = (2a - 2)/(3a + 6)
Cross multiplying,
2a (3a + 6) = 4a (2a - 2)
3a + 6 = 2 (2a - 2)
3a + 6 = 4a - 4
4a - 3a = 6 + 4
a = 10
VS = 2a = 20 units.
Hope this Helps!!!
(a) Both conditions are satisfied with <em>x</em> = (1, 0) for and <em>x</em> = (1, 0, 0) for
:
||(1, 0)|| = √(1² + 0²) = 1
max{1, 0} = 1
||(1, 0, 0)|| = √(1² + 0² + 0²) = 1
max{1, 0, 0} = 1
(b) This is the well-known triangle inequality. Equality holds if one of <em>x</em> or <em>y</em> is the zero vector, or if <em>x</em> = <em>y</em>. For example, in , take <em>x</em> = (0, 0) and <em>y</em> = (1, 1). Then
||<em>x</em> + <em>y</em>|| = ||(0, 0) + (1, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2
||<em>x</em>|| + ||<em>y</em>|| = ||(0, 0)|| + ||(1, 1)|| = √(0² + 0²) + √(1² + 1²) = √2
The left side is strictly smaller if both vectors are non-zero and not equal. For example, if <em>x</em> = (1, 0) and <em>y</em> = (0, 1), then
||<em>x</em> + <em>y</em>|| = ||(1, 0) + (0, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2
||<em>x</em>|| + ||<em>y</em>|| = ||(1, 0)|| + ||(0, 1)|| = √(1² + 0²) + √(0² + 1²) = 2
and of course √2 < 2.
Similarly, in you can use <em>x</em> = (0, 0, 0) and <em>y</em> = (1, 1, 1) for the equality, and <em>x</em> = (1, 0, 0) and <em>y</em> = (0, 1, 0) for the inequality.
(c) Recall the dot product identity,
<em>x</em> • <em>y</em> = ||<em>x</em>|| ||<em>y</em>|| cos(<em>θ</em>),
where <em>θ</em> is the angle between the vectors <em>x</em> and <em>y</em>. Both sides are scalar, so taking the norm gives
||<em>x</em> • <em>y</em>|| = ||(||<em>x</em>|| ||<em>y</em>|| cos(<em>θ</em>)|| = ||<em>x</em>|| ||<em>y</em>|| |cos(<em>θ</em>)|
Suppose <em>x</em> = (0, 0) and <em>y</em> = (1, 1). Then
||<em>x</em> • <em>y</em>|| = |(0, 0) • (1, 1)| = 0
||<em>x</em>|| • ||<em>y</em>|| = ||(0, 0)|| • ||(1, 1)|| = 0 • √2 = 0
For the inequality, recall that cos(<em>θ</em>) is bounded between -1 and 1, so 0 ≤ |cos(<em>θ</em>)| ≤ 1, with |cos(<em>θ</em>)| = 0 if <em>x</em> and <em>y</em> are perpendicular to one another, and |cos(<em>θ</em>)| = 1 if <em>x</em> and <em>y</em> are (anti-)parallel. You get everything in between for any acute angle <em>θ</em>. So take <em>x</em> = (1, 0) and <em>y</em> = (1, 1). Then
||<em>x</em> • <em>y</em>|| = |(1, 0) • (1, 1)| = |1| = 1
||<em>x</em>|| • ||<em>y</em>|| = ||(1, 0)|| • ||(1, 1)|| = 1 • √2 = √2
In , you can use the vectors <em>x</em> = (1, 0, 0) and <em>y</em> = (1, 1, 1).