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2 years ago

A dog walks at a speed of 3mi/hr for 2 hours. How far (distance) did the dog walk? *

Mathematics

1 answer:

Nikolay [14]2 years ago

6 0

Answer:

6 miles

Step-by-step explanation:

After one hour the dog will have walked three miles after two hours the dog will have walked three more miles for a total of 6 miles.

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Binomial Expansion/Pascal's triangle. Please help with all of number 5.

Mandarinka [93]

$\begin{matrix}1\\1&1\\1&2&1\\1&3&3&1\\1&4&6&4&1\end{bmatrix}$

The rows add up to $1,2,4,8,16$ , respectively. (Notice they're all powers of 2)

The sum of the numbers in row $n$ is $2^{n-1}$ .

The last problem can be solved with the binomial theorem, but I'll assume you don't take that for granted. You can prove this claim by induction. When $n=1$ ,

$(1+x)^1=1+x=\dbinom10+\dbinom11x$

so the base case holds. Assume the claim holds for $n=k$ , so that

$(1+x)^k=\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^{k-1}+\dbinom kkx^k$

Use this to show that it holds for $n=k+1$ .

$(1+x)^{k+1}=(1+x)(1+x)^k$
$(1+x)^{k+1}=(1+x)\left(\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^{k-1}+\dbinom kkx^k\right)$
$(1+x)^{k+1}=1+\left(\dbinom k0+\dbinom k1\right)x+\left(\dbinom k1+\dbinom k2\right)x^2+\cdots+\left(\dbinom k{k-2}+\dbinom k{k-1}\right)x^{k-1}+\left(\dbinom k{k-1}+\dbinom kk\right)x^k+x^{k+1}$

Notice that

$\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!}{\ell!(k-\ell)!}+\dfrac{k!}{(\ell+1)!(k-\ell-1)!}$
$\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(\ell+1)}{(\ell+1)!(k-\ell)!}+\dfrac{k!(k-\ell)}{(\ell+1)!(k-\ell)!}$
$\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(\ell+1)+k!(k-\ell)}{(\ell+1)!(k-\ell)!}$
$\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(k+1)}{(\ell+1)!(k-\ell)!}$
$\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{(k+1)!}{(\ell+1)!((k+1)-(\ell+1))!}$
$\dbinom k\ell+\dbinom k{\ell+1}=\dbinom{k+1}{\ell+1}$

So you can write the expansion for $n=k+1$ as

$(1+x)^{k+1}=1+\dbinom{k+1}1x+\dbinom{k+1}2x^2+\cdots+\dbinom{k+1}{k-1}x^{k-1}+\dbinom{k+1}kx^k+x^{k+1}$

and since $\dbinom{k+1}0=\dbinom{k+1}{k+1}=1$ , you have

$(1+x)^{k+1}=\dbinom{k+1}0+\dbinom{k+1}1x+\cdots+\dbinom{k+1}kx^k+\dbinom{k+1}{k+1}x^{k+1}$

and so the claim holds for $n=k+1$ , thus proving the claim overall that

$(1+x)^n=\dbinom n0+\dbinom n1x+\cdots+\dbinom n{n-1}x^{n-1}+\dbinom nnx^n$

Setting $x=1$ gives

$(1+1)^n=\dbinom n0+\dbinom n1+\cdots+\dbinom n{n-1}+\dbinom nn=2^n$

which agrees with the result obtained for part (c).

4 0

3 years ago

What is another way to write the absolute value inequality p ≤ 12? O-12≤p≤ 12 O-122p≤12 O-12 ≤por p≤ 12 -122P or p≤ 12 958 What

victus00 [196]

Answer:

-12 ≤ p ≤ 12

Step-by-step explanation:

If the absolute value inequality is

|p| ≤ 12,

then the answer is

-12 ≤ p ≤ 12

4 0

2 years ago

The base of the right prism is a rhombus with the side 7 cm. The altitude of the prism is h=13 cm. Find the lateral area of the

Art [367]

Answer:The lateral area is $364\ cm^{2}$

Step-by-step explanation:

we know that

The lateral area of the right prism is equal to

$LA=Ph$

where

P is the perimeter of the base

H is the height of the prism

Find the perimeter of the base P

$P=4(7)=28\ cm$ ----> the rhombus has four equal sides

$h=13\ cm$

substitute

$LA=28(13)=364\ cm^{2}$

6 0

3 years ago

Read 2 more answers

Please help me with these two math questions! Thank you ! :)

notka56 [123]

I think it might be c hope that helps you

5 0

2 years ago

Mai has to decide between two cafeteria meal plans. Under plan A, each meal cost $2.50. Under plan B, one month of meals cost $3

BaLLatris [955]

Answer:

If he eats more then 12 lunches in a month he should take Plan B, if he eats less then 12 lunches a month he should take Plan A.

Step-by-step explanation:

2.5x12=30

If he is on Plan A and buys 13 lunches it would cost more money then to be on Plan B. If he buys less then it would cost him less money to be on Plan A since he would be paying under $30.

4 0

2 years ago

A dog walks at a speed of 3mi/hr for 2 hours. How far (distance) did the dog walk? *​

A dog walks at a speed of 3mi/hr for 2 hours. How far (distance) did the dog walk? *