We are given with the rate of change of the base, db/dt equal to 1 ft/ sec and the rate of change of the height of the triangle, dh/dt equal to 2 ft/sec. b is 10 ft and h is 70 ft. Area of triangle is equal to A= 0.5 bh The rate of change of the area is equal to dA/dt = 0.5 b dh/dt + 0.5 h db/dt. Substituting, dA= 0.5*10*2 + 0.5*70 * 1 equal to 45 ft2 / sec.

3 0

3 years ago

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Angelica painted 1/20 of her bedroom walls in 1/2 an hour. How long will it take Angelica to paint her whole bedroom?

julsineya [31]

Angelica would have painted the bedroom in 20 hours if every hour she painted 1/20. But she doesn't paint 1/20 of a room per hour but rather 30 minutes. That means it will take her 10 hours to paint the whole room.

Hope this helps.

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3 years ago

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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} }"

Feliz [49]

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,

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

where the $c_i$ are positive integer coefficients from the binomial expansion. By Euler's theorem,

$\left(32^{(\cdots)\right)^6 \equiv 1 \pmod9$

so that

$32^{37^{32}} \equiv 32\cdot1 \equiv \boxed{5} \pmod9$

7 0

1 year ago