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EastWind [94]
2 years ago
12

A rectangular rug has an area of 84 square feet. If the length is 8 feet more than the width, find the dimensions of the rug.​

Mathematics
1 answer:
Alex17521 [72]2 years ago
8 0

Answer:

w=6

l=14

Step-by-step explanation:

Area of rectangle = l*w

So:

w*(w+8)=84

Then, just find w, and w+8.

w^2+8w=84\\w^2+8w-84=0\\(w-6)(w+14)=0\\w-6=0, w=6\\w+14=0, w=-14

Length cannot be negative, so the width is 6.

Length = width + 8 = 6+8=14

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How many sets of three consecutive integers are there in which the sum of the three integers equals their product?
skad [1K]

Answer:

3

Step-by-step explanation:

since the 3 integers are consecutive, we are dealing with x, x+1, x+2.

and their sum is the same as their product :

x + (x + 1) + (x + 2) = x(x + 1)(x + 2)

3x + 3 = x(x² + 3x + 2) = x³ + 3x² + 2x

x³ + 3x² - x - 3 = 0

this is a polynomial of third degree.

and as such it has 3 solutions.

of course, it could be that some of them are the same or are even in the realm of complex numbers (i = sqrt(-1)), but usually these 3 solutions are different real numbers.

I tried x=1 just to see, and, hey, it is a solution for this equation.

x = 1 means that the other 2 consecutive integers are 2 and 3.

and indeed, 1+2+3 = 1×2×3 = 6.

now it is easier to find the other 2 solutions, as a zero solution can be expressed as a factor of the whole expression.

for x = 1 the factor term is (x - 1), as this term is then turning 0, when x = 1.

I can divide the main expression by this factor and then analyze the quotient about the other 2 solutions.

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- x³ - x²

----------------

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0 0

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(x² + 4x + 3)(x - 1).

now we need to find the 2 zero solutions for x²+4x+3

the general solution to a quadratic equation is

x = (-b ± sqrt(b² - 4ac))/(2a)

in our case

a = 1

b = 4

c = 3

so,

x = (-4 ± sqrt(4² - 4×1×3))/(2×1) =

= (-4 ± sqrt(16 - 12))/2 = (-4 ± sqrt(4))/2 =

= (-4 ± 2)/2 = -2 ± 1

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and there we have it fully proven :

there are 3 different sets of 3 consecutive integers with the same sum as product.

4 0
2 years ago
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Brilliant_brown [7]

Answer:48

Step-by-step explanation:

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The correct answer should be A.
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Step-by-step explanation:

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