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3 years ago

6

Write an integers to represent each situation. 1.An $80 withdrawal. 2. A 2-inch in rise in water level 3. A lose of 15 pounds. 4

. Draining 15 gallons of water from a fish tank.

1 answer:

Orlov [11]3 years ago

6 0

Answer:

-80 ; 2 ; - 15 ; - 15

Step-by-step explanation:

Given that:

1.)

A withdrawal of $80, will cause reduction in the initial value, hence, negative

= - 80

2.)

A 2 - inch rise in water level, indicates increase from initial level, hence, positive

= + 2

3.)

A lose of 15 pounds indicates reduction in weight, hence, negative

= - 15

4.)

Draining 15 gallons of water from a fish tank indicates reduction Hence, negative

= - 15

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Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$

Viktor [21]

When $n=0$ , we have

$5^{3^0} + 1 = 5^1 + 1 = 6$

$3^{0 + 1} = 3^1 = 3$

and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)

Suppose this is true for $n=k$ , that

$3^{k + 1} \mid 5^{3^k} + 1$

Now for $n=k+1$ , we have

$5^{3^{k+1}} + 1 = 5^{3^k \times 3} + 1 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k}\right)^3 + 1^3 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k} + 1\right) \left(\left(5^{3^k}\right)^2 - 5^{3^k} + 1\right)$

so we know the left side is at least divisible by $3^{k+1}$ by our assumption.

It remains to show that

$3 \mid \left(5^{3^k}\right)^2 - 5^{3^k} + 1$

which is easily done with Fermat's little theorem. It says

$a^p \equiv a \pmod p$

where $p$ is prime and $a$ is any integer. Then for any positive integer $x$ ,

$5^3 \equiv 5 \pmod 3 \implies (5^3)^x \equiv 5^x \pmod 3$

Furthermore,

$5^{3^k} \equiv 5^{3\times3^{k-1}} \equiv \left(5^{3^{k-1}}\right)^3 \equiv 5^{3^{k-1}} \pmod 3$

which goes all the way down to

$5^{3^k} \equiv 5 \pmod 3$

So, we find that

$\left(5^{3^k}\right)^2 - 5^{3^k} + 1 \equiv 5^2 - 5 + 1 \equiv 21 \equiv 0 \pmod3$

QED

5 0

2 years ago

What number makes the expressions equivalent? Enter your answer in the box. 1/2(–1.4m + 0.4) =__m + 0.2A) -1.4B) 1.4C) -0.7D) 0.

Snowcat [4.5K]

Answer:

C. -0.7

Explanation:

Given the equation:

$\frac{1}{2}(-1.4m+0.4)=\boxed{\square}_{}m+0.2$

First, distribute the bracket on the left-hand side:

$\begin{gathered} \frac{1}{2}(-1.4m)+\frac{1}{2}(0.4) \\ =-0.7m+0.2 \end{gathered}$

The number that makes the given expressions equivalent is -0.7.

The correct choice is C.

6 0

1 year ago

If the rate of inflation is 3.4%per year, the future price pt(in dollars) of a certain item can be modeled by the following expo

Doss [256]

Answer:

The current price is $3000 and price after 9 years from today is $4054.

Step-by-step explanation:

The future price pt(in dollars) of a certain item can be modeled by the following exponential function

$p(t)=3000(1.034)^t$

where, t is the number of years from today.

Substitute t=0 to find the current price.

$p(0)=3000(1.034)^0=3000$

Therefore the current price is $3000.

Substitute t=9 to find the price after 9 years from today.

$p(9)=3000(1.034)^9$

$p(9)=3000(1.35109177087)$

$p(9)=4053.27531261$

$p(9)\approx 4054$

Therefore the price after 9 years from today is $4054.

4 0

3 years ago

The distance from first base to third base is 1.4times the distance from home plate to third base, which is k feet. REPRESENT TH

geniusboy [140]

Answer:

1.4 times k

Step-by-step explanation:

If k feet is the distance from home plate to third base, and the distance from first base to third base is 1.4 times the previous value, then

distance from first base to third base = 1.4 * k or 1.4 times k

If, for example, k=90 feet, then from first to third base there are

1.4*90=126 feet

4 0

3 years ago

What is the equation for the line that passes through the points (-2,1) and (1,-11)

givi [52]

Answer:

y=4x+7

Step-by-step

(-2,1) and (1,-11)

1)Rule you should use is run over rise aka slope

y²-y^1=

x^2-x^1=

(-11)-1=12

1-(-2)=3

12/3=4

slope =4

2)Rule of point slope intercept

y - y1 = m(x - x1)

y+1=4(x+2)

y+1=4x+8

-1 -1

y=4x+7

8 0

3 years ago