Answer:
The dimension of the rectangular closet before construction = (x + 5) ft by (x + 4) ft
Step-by-step explanation:
The homeowner has a rectangular closet and the area of the closet is (x² + 9x + 20) ft². The area of a rectangle is the length multiplied by the width. The length is given as (x + 5) ft. After construction the area change to (x² + 14x + 48) ft². The length became (x + 6) ft.
The dimensions before construction can be calculated as follows:
Mathematically,
area of rectangle = Length × width
area = (x² + 9x + 20) ft².
Length = (x + 5) ft
Width = unknown
area = LW
(x² + 9x + 20) = (x + 5) W
W = (x² + 9x + 20) / x + 5
W = (x + 4) ft
The dimension of the rectangular closet before construction = (x + 5) ft by (x + 4) ft
<span>1. Find a ⋅ b.
a = <7, 4>, b = <3, 5> (2 points)
41 <---------- right answer
<10, 9>
-1
<21, 20>
Explanation:
a . b means the dot or scalar product of the vectors a and b.
The scalar product of vectorr <x1,y2> and <x2,y2> is (x1)*(y1) + (x2)*(y2)
So, a . b = 7*3 + 4*5 = 21 + 20 = 41 <-------- answer
2. Find a ⋅ b
a = 10i + 9j, b = 4i + 3j (2 points)
<40, 27>
<14, 12>
67 <---------- answer
-13
Explanation:
Again you are required to find the dot product of two vectors. In this case the vectors are given using the representation with the unit vectors i and j.
The dot product of the vectors (x1 i + y1 j) . (x2 i + y2 j) is x1*x2 + y1*y2
So, (</span><span><span>10i + 9j) . (4i + 3j) = 10*4 + 9*3 = 40 + 27 = 67
Answer: 67
</span>
3. Find the angle between the given vectors to the nearest tenth of a degree.
u = <6, -1>, v = <7, -4> (2 points)
20.3° <--------------- answer
10.2°
0.2°
30.3°
Explanation:
You can use the dot product to find the angle between two vectors.
This is the formula:
cos(α) = [ <a> dot <b. ] / [|a| |b| ]
where α is the angle between the vector a and b.
<a> is the vector a, <b> is the vector b, <a> dot <b> is the dot product, |a| is the magnitude of vector a, and |b| is the magnitude of vector b.
<u> dot <v> = (6)(7) + (-1)(-4) = 42 + 4 = 46
|u| = √ (6^2 + 1^2) = √37
|v| = √ (7^2 + 4^2) = √65
=> cos(α) = [46] / [√37 * √65] = 0.938
=> α = arccos(0.938) = 20.3°
4. Determine whether the vectors u and v are parallel, orthogonal, or neither.
u = <10, 6>, v = <9, 5> (2 points)
Orthogonal
Parallel
Neither <------------ answer
Explanation:
At sight they are neither orthogonal nor parallel. You can put the points in a graph and you will realize inmediately.
Let's calculate the angle, with the formula given in the above problem.
cos(α) = [<u> dot< >v] / [ |u| |v| ]
<u> dot <v> = 10*9 + 6*5 = 90 + 30 = 120
|u| = √(10^2 + 6^2) = √136
|v| = √(9^2 + 5^2) = √106
cos(α) = 120 / (√136 * √106] = 0,999
α = arctan(0,999) = 1,9°
Which confirms that they are neither orthogonal nor parallel
5. Evaluate the expression:
v ⋅ w
Given the vectors:
r = <8, 8, -6>; v = <3, -8, -3>; w = <-4, -2, -6> (2 points)
</span>
Solution:
v . w = v dot w = (3)(-4) + (-8)(-2) + (-3)(-6) = -12 + 16 + 18 = 22
Answer: 22
Answer:
Step-by-step explanation:
I'm not sure if the right side is 
or 
, so I'll provide an answer for both:
Answer:
106.08 and 47.13 feet
Step-by-step explanation:
We have that the area of a rectangle is equal to the multiplication of one side by the other side, that is:
a = x * y
we know that the area is 5000, therefore:
5000 = x * y
y = 5000 / x
they tell us that the diagonal measures 10 feet more than one side, let's say on the x side, therefore the diagonal measures x + 10.
A right triangle is formed, which by the Pythagorean equation has the following form:
(x + 10) ^ 2 = x ^ 2 + y ^ 2
if we replace the value of and we are left with:
(x + 10) ^ 2 = x ^ 2 + (5000 / x) ^ 2
x ^ 2 + 20 * x + 100 = x ^ 2 + 5000 ^ 2 / x ^ 2
20 * x + 100 = 5000 ^ 2 / x ^ 2
20 * x ^ 3 + 100 * x ^ 2 = 25000000
from here we obtain that the value of x is equal to 106.08
now the value of y:
y = 5000 / 106.08
y = 47.13
therefore the measurements are 106.08 and 47.13 feet
Answer:
6i square root of 2 or 8.48528137 i
Step-by-step explanation:
