Find the volume of the figure

Mathematics

2 answers:

Nadusha1986 [10]3 years ago

8 0

Just find the base area then multiply it by the height.
so here for the base area we can divide it in 2 parts 6x6 + 2x1
which the total is 38
then multiply 38 by 1, which is the height, so it's 38

HACTEHA [7]3 years ago

6 0

38(inches cubed)
V=Bh
B=area of the base
split into two
B=1*2=1 H=1 V=2*1 V=2

B=6*6=36 H=1 V=36*1 V=36

36+2=38

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WHAT IS THE REMAINDER WHEN <img src="https://tex.z-dn.net/?f=32%5E%7B37%5E%7B32%7D%20%7D" id="TexFormula1" title="32^{37^{32} }"

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Recall Euler's theorem: if $\gcd(a,n) = 1$ , then

$a^{\phi(n)} \equiv 1 \pmod n$

where $\phi$ is Euler's totient function.

We have $\gcd(9,32) = 1$ - in fact, $\gcd(9,32^k)=1$ for any $k\in\Bbb N$ since $9=3^2$ and $32=2^5$ share no common divisors - as well as $\phi(9) = 6$ .

Now,

$37^{32} = (1 + 36)^{32} \\\\ ~~~~~~~~ = 1 + 36c_1 + 36^2c_2 + 36^3c_3+\cdots+36^{32}c_{32} \\\\ ~~~~~~~~ = 1 + 6 \left(6c_1 + 6^3c_2 + 6^5c_3 + \cdots + 6^{63}c_{32}\right) \\\\ \implies 32^{37^{32}} = 32^{1 + 6(\cdots)} = 32\cdot\left(32^{(\cdots)}\right)^6$