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Hatshy [7]
2 years ago
8

How do I convert these probs? into base 10 4120_7, A3B_12, WXYZ_36

Mathematics
1 answer:
natali 33 [55]2 years ago
7 0

4120_7 = 4•7³ + 1•7² + 2•7¹ + 0•7⁰

4120_7 = 4•343 + 1•49 + 2•7 + 0•1

4120_7 = 1372 + 49 + 14

4120_7 = 1435

In base 12, we use the digits 0-9 as well as A for 10 and B for 11. So

A3B_12 = 10•12² + 3•12¹ + 11•12⁰

A3B_12 = 10•144 + 3•12 + 11•1

A3B_12 = 1440 + 36 + 11

A3B_12 = 1487

In base 36, we assign values between 10 and 35 to the letters A-Z, so that

WXYZ_36 = 32•36³ + 33•36² + 34•36¹ + 35•36⁰

WXYZ_36 = 32•46656 + 33•1296 + 34•36 + 35•1

WXYZ_36 = 1492992 + 42768 + 1224 + 35

WXYZ_36 = 1537019

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So, Bob needs 2 cups of concentrate (I'll use the variable c to shorten it) for every 3.5 cups of water (w). When he has 6 cups of c, he has three times the original recipe, and therefore we need 3w as well. 

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2. f(x)=\dfrac18\left(\dfrac14\right)^{x-2}

Domain: (-\infty,\infty), because any value of x is allowed and gives a number f(x).

Range: (0,\infty), because a^x>0 for any positive real a\neq0.

y-intercept: This is a point of the form (0, f(0)). So plug in x=0; we get f(0)=\frac18\left(\frac14\right)^{-2}=\frac{4^2}8=2. So the intercept is (0, 2), or just 2. (Interestingly, you didn't get marked wrong for that...)

Asymptote: This can be deduced from the range; the asymptote is the line y=0.

Increasing interval: Going from left to right, there is no interval on which f(x) is increasing, since 1/4 is between 0 and 1.

Decreasing interval: Same as the domain; f(x) is decreasing over the entire real line.

End behavior: The range tells you f(x)>0, and you know f(x) is decreasing over its entire domain. This means that f(x)\to+\infty as x\to-\infty, and f(x)\to0 and x\to+\infty.

3. f(x)=\left(\dfrac32\right)^{-x}-7

Domain: Same as (2), (-\infty, \infty).

Range: We can rewrite f(x)=\left(\frac23\right)^x-7. \left(\frac23\right)^x>0 for all x, so \left(\frac23\right)^x-7>-7 for all x. Then the range is (-7,\infty).

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Asymptote: y=-7

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Decreasing interval: (-\infty,\infty)

End behavior: Similar to (2), but this time f(x)\to+\infty as x\to-\infty and f(x)\to-7 as x\to+\infty.

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