Remember the chain rule.
L(x)=f(g(x))
L'(x)=f'(g(x))g'(x)
take the derivative of f(g(x)). just treat them like they are variables. so you get:
h'=f'(g(x))g'(x)
now plug in your x value and evaluate:
h'(1)=f'(g(1))(g'(1))
substitute in values that you know and evaluate again
h'(1)=f'(3)(-3)
h'(1)=(-5)(-3)=15
Answer: Choice B) {3, 5, sqrt(34)}
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Explanation:
We can only have a right triangle if and only if a^2+b^2 = c^2 is a true equation. The 'c' is the longest side, aka hypotenuse. The legs 'a' and 'b' can be in any order you want.
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For choice A,
a = 2
b = 3
c = sqrt(10)
So,
a^2+b^2 = 2^2+3^2 = 4+9 = 13
but
c^2 = (sqrt(10))^2 = 10
which is not equal to 13 from above. Cross choice A off the list.
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Checking choice B
a = 3
b = 5
c = sqrt(34)
Square each equation
a^2 = 3^2 = 9
b^2 = 5^2 = 25
c^2 = (sqrt(34))^2 = 34
We can see that
a^2+b^2 = 9+25 = 34
which is exactly equal to c^2 above. This confirms the answer.
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Let's check choice C
a = 5, b = 8, c = 12
a^2 = 25, b^2 = 64, c^2 = 144
So,
a^2+b^2 = c^2
25+64 = 144
89 = 144
which is a false equation allowing us to cross choice C off the list.
We know that,
Julia can finish a 20-mile bike ride in 1.2 hours.
The distance Julia travels is 20 miles and the time she takes is 1.2 hours.
So, Julia's speed =
= 16.67 mph
Katie can finish the same bike ride in 1.6 hours.
The distance Katie travels is 20 miles and the time she takes is 1.6 hours.
So, Katie's speed =
= 12.5 mph
Now, to find how much faster Julia rides than Katie we subtract Katie's speed from Julia's speed.
So, 16.67 mph - 12.5 mph = 4.17 mph = 4.2 mph (approximately)
Thus, Julia rides 4.2 mph faster than Katie.