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3 years ago

Find the dimensions of a rectangle, length 7m less than three times the width and the area of the rectangle is 66m^2.

Mathematics

1 answer:

zhuklara [117]3 years ago

5 0

We know: length of a rectangle is 7m less than three times the width.

So let's set the length to l, width to w.

And we can get that l = 3w-7.

We also know that the area of the rectangle is 66 m^2.

Thus, l*w= 66.

So we have:

function 1: l=3w-7

function 2: l*w=66

plug function 1 to 2.

(3w-7)*w=66

3w^2-7w-66=0

w=6 and -11/3.

And the width of a rectangle can not be negative.

So: w=6.

Plug w=6 back to function 1 or 2, either way is correct and can give you the length. In this case, I will plug it back to function 1.

l=3*6-7=11.

So: the width of the rectangle is 6, and the length of the rectangle is 11.

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How do you solve x for (x+5)^3/2 = ( x-1)^3

malfutka [58]

Answer:

x = 6 (2^(1/3) + 2^(2/3)) + 7 or x = 6 (-2)^(1/3) ((-1)^(1/3) - 2^(1/3)) + 7 or x = 6 (-2)^(1/3) ((-2)^(1/3) - 1) + 7

Step-by-step explanation:

Solve for x:

1/2 (x + 5)^3 = (x - 1)^3

Expand out terms of the right hand side:

1/2 (x + 5)^3 = x^3 - 3 x^2 + 3 x - 1

Subtract x^3 - 3 x^2 + 3 x - 1 from both sides:

1 - 3 x + 3 x^2 - x^3 + 1/2 (x + 5)^3 = 0

Expand out terms of the left hand side:

-x^3/2 + (21 x^2)/2 + (69 x)/2 + 127/2 = 0

Bring -x^3/2 + (21 x^2)/2 + (69 x)/2 + 127/2 together using the common denominator 2:

1/2 (-x^3 + 21 x^2 + 69 x + 127) = 0

Multiply both sides by 2:

-x^3 + 21 x^2 + 69 x + 127 = 0

Multiply both sides by -1:

x^3 - 21 x^2 - 69 x - 127 = 0

Eliminate the quadratic term by substituting y = x - 7:

-127 - 69 (y + 7) - 21 (y + 7)^2 + (y + 7)^3 = 0

Expand out terms of the left hand side:

y^3 - 216 y - 1296 = 0

Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:

-1296 - 216 (z + λ/z) + (z + λ/z)^3 = 0

Multiply both sides by z^3 and collect in terms of z:

z^6 + z^4 (3 λ - 216) - 1296 z^3 + z^2 (3 λ^2 - 216 λ) + λ^3 = 0

Substitute λ = 72 and then u = z^3, yielding a quadratic equation in the variable u:

u^2 - 1296 u + 373248 = 0

Find the positive solution to the quadratic equation:

u = 864

Substitute back for u = z^3:

z^3 = 864

Taking cube roots gives 6 2^(2/3) times the third roots of unity:

z = 6 2^(2/3) or z = -6 (-1)^(1/3) 2^(2/3) or z = 6 (-2)^(2/3)

Substitute each value of z into y = z + 72/z:

y = 6 2^(1/3) + 6 2^(2/3) or y = 6 (-1)^(2/3) 2^(1/3) - 6 (-1)^(1/3) 2^(2/3) or y = 6 (-2)^(2/3) - 6 (-2)^(1/3)

Bring each solution to a common denominator and simplify:

y = 6 (2^(1/3) + 2^(2/3)) or y = 6 (-2)^(1/3) ((-1)^(1/3) - 2^(1/3)) or y = 6 (-2)^(1/3) ((-2)^(1/3) - 1)

Substitute back for x = y + 7:

Answer: x = 6 (2^(1/3) + 2^(2/3)) + 7 or x = 6 (-2)^(1/3) ((-1)^(1/3) - 2^(1/3)) + 7 or x = 6 (-2)^(1/3) ((-2)^(1/3) - 1) + 7

7 0

3 years ago

Read 2 more answers

You roll two fair dice, a green one and a red one. (a) What is the probability of getting a sum of 6? (Enter your answer as a fr

Stels [109]

Answer:

5/36 ; 1/12 ; 2/9 ; yes

Step-by-step explanation:

Given the following :

Roll of two fair dice : green and red

Probability = (number of required outcomes / number of total possible outcomes)

(a) What is the probability of getting a sum of 6?

Number of required outcomes = 5

P(sum of 6) = 5/36

b.) What is the probability of getting a sum of 10?

Number of required outcomes = 3

P(sum of 10) = 3 / 36 = 1/12

c.) What is the probability of getting a sum of 6 or 10?

P(getting a sum of 6) + P(getting a sum of 10)

(5/36) + (1/12) = (5 + 3) / 36

= 8/36 = 2/9

The events are mutually exclusive because each event cannot occur at the same time.

8 0

4 years ago

Use the roster method to write the given set the set of integers x that satisfy 2x-1= -13

anygoal [31]

The only set of elements that satisfied the equation 2x-1= -13 is {-6}.

A set is a combination of specific quantities in which the meaning of each variable must be the same.

For example {1,2,3,4,5,6...} is a set of natural numbers in which each variable is representing a natural number so the overall meaning is the same among all.

Another example could be a set of whole numbers like that.

Given the equation,

2x-1= -13

By taking -1 right-hand side

2x = -13 + 1 = -12

By dividing 2

x = -6

Hence "The only set of elements that satisfied the equation 2x-1= -13 is {-6}".

To learn more about the sets,

brainly.com/question/8053622

#SPJ4

6 0

1 year ago

What is the 20th term of 64,67,70,73,76

PIT_PIT [208]

Let's analyze the increments for each step of the sequence:

Each step we add 3 to the previous number.

Since we need the 20th, from what we saw, in the 20th term we will have added three 20 times.

So the procedure is: calculate how much is 3 times 20 and then add that to the first term of the sequence.

$3\times20=60$

So we add 60 to the first term to find the 20th term:

$64+60=124$

Answer: 124

6 0

1 year ago

HELP ME GUYS

Tatiana [17]

Answer:

aida's work is the correct answer

7 0

2 years ago