Mathematics, 28.06.2019 14:30 cadenm81

The area of a rectangle is 27 square meters. if the length is 6 meters less than 3 times the width, then find the dimensions of the rectangle. round off your answers to the nearest hundredth

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Answer from: caldonjoshhsms2061

width = 4.16 meter

length = 6.49 meter

Step-by-step explanation:

Area of the rectangle =27 m²

Let the width of the rectangle be x meter

So, Length = 3 * width - 6

= 3*x - 6

= 3x-6 meter

Area of the rectangle = length * width

27 = (3x-6)*x

Flipping the sides of the equation, we have

(3x-6)*x =27

Distributing the left side, we get

(3x)*(x) - (6)*(x) = 27

=> 3x² - 6x = 27

Subtract 27 from both sides,

3x² - 6x -27 = 27 - 27

=> 3x² - 6x -27 = 0

Factoring out 3 from all the terms on the left side, we have

3(x² - 2x -9) = 0

Dividing both sides by 3, we have

$\frac{3(x^{2}-2x-9) }{3}$ = $\frac{0}{3}$

Cancelling out the 3's on the left, we get

x² - 2x -9 = 0

We'll use the quadratic formula to solve for the x,

x = $\frac{-b\pm\sqrt{b^{2}-4ac } }{2a}$

Comparing the quadratic equation x² - 2x -9 = 0 with ax² + bx + c = 0, we get

a = 1 (as x² has no coefficient)

b = -2

c = -9

Plugging in the values of a, b, and c into the quadratic formula, we get

x = $\frac{-(-2)\pm\sqrt{(-2)^{2}-4(1)(-9) } }{2(1)}$

=> x = $\frac{2\pm\sqrt{4+36 } }{2}$

=> x = $\frac{2\pm\sqrt{40}}{2}$

=> x = $\frac{2\pm2\sqrt{10}}{2}$

Factoring out 2 from the top, we get

x = $\frac{2(1\pm\sqrt{10})}{2}$

Canceling out the 2's from the top and bottom, we have

x = $1\pm\sqrt{10}$

Either x = $1+\sqrt10$ or x= $1-\sqrt10$

=> x = 1 + 3.162 or x = 1 - 3.162

=> x = 4.162 (possible) or x = -2.162 (not possible as width can't be negative)

So, width = 4.16 meter (rounded off to the nearest hundredth)

Now,

Area of the rectangle = length * width

27 = length * 4.16

Flipping the sides of the equation,

length * 4.16 = 27

Dividing both sides by 4.16, we get

$\frac{length * 4.16}{4.16} = \frac{27}{4.16}$

Cancelling out 4.16 from the top and bottom of the left side, we get

length = 6.490

=> length = 6.49 meter (rounded off to the nearest hundredth)

Answer from: Quest

answer are

step-by-step explanation:

Answer from: Quest

$a_1 = 1$ , and

$a_{n+1} = a_n + (n + 1)$ for $n \ge 1$

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote $a_1 = 1$ , $a_2 = 3$ , $a_3 = 6$ , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from $a_1 = 1$ , to get to $a_2 = 3$ , we add $2(=1+1)$

from $a_2 = 3$ , to get to $a_3 = 6$ , we add $3 (=2+1)$

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

$a_1 = 1$ , and $a_{n+1} = a_n +(n + 1)$ for $n \ge 1$

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