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1 year ago

two rectangles have the exact same area. the measurements of the first rectangle is 3k by 3.calculate the value of k

Mathematics

1 answer:

photoshop1234 [79]1 year ago

5 0

The value of k is 6.

Given we have length as 3k and the width as 3 of the first rectangle.

length = k₊3 and width = 6 of the second recatngle

Two rectangles has the same exact area.

area = length × breadth

3k × 3 = (k₊3)6

9k = 6k ₊ 18

3k = 18

k = 18/3

k = 6

hence we get the value of k as 6.

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Prove the following by induction. In each case, n is apositive integer.<br> 2^n ≤ 2^n+1 - 2^n-1 -1.

frutty [35]

<h2>Answer with explanation:</h2>

We are asked to prove by the method of mathematical induction that:

$2^n\leq 2^{n+1}-2^{n-1}-1$

where n is a positive integer.

Let us take n=1

then we have:

$2^1\leq 2^{1+1}-2^{1-1}-1\\\\i.e.\\\\2\leq 2^2-2^{0}-1\\\\i.e.\\2\leq 4-1-1\\\\i.e.\\\\2\leq 4-2\\\\i.e.\\\\2\leq 2$

Hence, the result is true for n=1.

Let us assume that the result is true for n=k

i.e.

$2^k\leq 2^{k+1}-2^{k-1}-1$

Now, we have to prove the result for n=k+1

i.e.

<u>To prove:</u> $2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-1$

Let us take n=k+1

Hence, we have:

$2^{k+1}=2^k\cdot 2\\\\i.e.\\\\2^{k+1}\leq 2\cdot (2^{k+1}-2^{k-1}-1)$

( Since, the result was true for n=k )

Hence, we have:

$2^{k+1}\leq 2^{k+1}\cdot 2-2^{k-1}\cdot 2-2\cdot 1\\\\i.e.\\\\2^{k+1}\leq 2^{(k+1)+1}-2^{k-1+1}-2\\\\i.e.\\\\2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-2$

Also, we know that:

$-2$

(

Since, for n=k+1 being a positive integer we have:

$2^{(k+1)+1}-2^{(k+1)-1}>0$ )

Hence, we have finally,

$2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-1$

Hence, the result holds true for n=k+1

Hence, we may infer that the result is true for all n belonging to positive integer.

i.e.

$2^n\leq 2^{n+1}-2^{n-1}-1$ where n is a positive integer.

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lara31 [8.8K]

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The cost of a ticket to the circus is $14.00 for children and $32.00 for adults. On a certain day, attendance at the circus was

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400 children and 600 adults bought tickets.

Step-by-step explanation:

Given,

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Let,

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y be the number of adults.

According to given statement;

x+y=1000 Eqn 1

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Multiplying Eqn 1 by 14

$14(x+y=1000)\\14x+14y=14000\ \ \ Eqn\ 3$

Subtracting Eqn 3 from Eqn 2

$(14x+32y)-(14x+14y)=24800-14000\\14x+32y-14x-14y=10800\\18y=10800\\$

Dividing both sides by 18

$\frac{18y}{18}=\frac{10800}{18}\\y=600$

Putting y=800 in Eqn 1

$x+600=1000\\x=1000-600\\x=400$

400 children and 600 adults bought tickets.

Keywords: linear equations, subtraction

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3 years ago

two rectangles have the exact same area. the measurements of the first rectangle is 3k by 3.calculate the value of k​

two rectangles have the exact same area. the measurements of the first rectangle is 3k by 3.calculate the value of k