From the table, for every 6 containers of water, you need 8 containers of red dye. So, for every one container of red dye you need 0.75 containers of water (6/8). Then, multiply that by 100 (since you wanted to know water necessary for 100 containers of red dye). For 100 containers of red dye, you would need 75 containers of water.
Since f(x) is (strictly) increasing, we know that it is one-to-one and has an inverse f^(-1)(x). Then we can apply the inverse function theorem. Suppose f(a) = b and a = f^(-1)(b). By definition of inverse function, we have
f^(-1)(f(x)) = x
Differentiating with the chain rule gives
(f^(-1))'(f(x)) f'(x) = 1
so that
(f^(-1))'(f(x)) = 1/f'(x)
Let x = a; then
(f^(-1))'(f(a)) = 1/f'(a)
(f^(-1))'(b) = 1/f'(a)
In particular, we take a = 2 and b = 7; then
(f^(-1))'(7) = 1/f'(2) = 1/5
Answer:
<em>Gerry arrived at the bus station at 12:30 P.M.</em>
Step-by-step explanation:
From noon and 5:00 P.M. there are 5 hours.
Gerry, Dale, and Pat arrived at the bus stop within that interval, which means the sum of the times elapsed between their arrival times must be 5.
4 out of the 5 hours elapsed since Gerry and Dale's arrivals. This means there is only one hour left for twice x.
If twice x is one hour, then x is half an hour. Thus, the complete sequence of arrivals is:
Gerry arrived at the bus station at 12:30 P.M.
Dale arrived at the bus station at 4:30 P.M.
Pat arrived at the bus station at 5:00 P.M.
Answer:
A Between 1.43 in. and 1.47 in.
Step-by-step explanation:
If the circumference is between 9 and 9.25 inches, and we know that circumference equals 2 * pi * radius, we can solve to find that the radius equals the circumference divided by 2pi. So we get between 1.43 in and 1.47in
A complex mathematical topic, the asymptotic behavior of sequences of random variables, or the behavior of indefinitely long sequences of random variables, has significant ramifications for the statistical analysis of data from large samples.
The asymptotic behavior of the sample estimators of the eigenvalues and eigenvectors of covariance matrices is examined in this claim. This work focuses on limited sample size scenarios where the number of accessible observations is comparable in magnitude to the observation dimension rather than usual high sample-size asymptotic .
Under the presumption that both the sample size and the observation dimension go to infinity while their quotient converges to a positive value, the asymptotic behavior of the conventional sample estimates is examined using methods from random matrix theory.
Closed form asymptotic expressions of these estimators are obtained, demonstrating the inconsistency of the conventional sample estimators in these asymptotic conditions, assuming that an asymptotic eigenvalue splitting condition is satisfied.
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