Geometric means A quantity grows exponentially according to y(...

Question

Geometric means A quantity grows exponentially according to y(t) = y₀eᵏᵗ. What is the relationship among m, n, and p such that y(p) = √(y(m)y(n))?

Accepted Answer

Welcome back, everyone. Let F of T in the form of a multiplied by E raises the power of C T describe the temperature of a substance over time. If F of S is equal to square root of F of U multiplied by F of V, what is the relationship among S, U, and V? Assume that A is greater than 0 and C is not equal to 0. A S equals U minus V divided by 2. B S equals U + V. C S equals UV divided by 2, and D S equals U + V divided by 2. So for this problem, let's identify each parameter, starting with FFS. We can show that it is equal to a multiplied by E raise to the power of CS. In other words, T is replaced with S. Now, similarly, F of U is going to be a multiplied by E, a race to the power of C U. And F of V is going to be equal to a multiplied by E or raise to the power of CV. We know that FOS, which is a. E to the power of C S is equal to square root of F of u multiplied by F V. So square root of a E to the power of Cu multiplied by Ae raises the power of CV. What we want to do is simplify. We can show that a multiplied by E to the power of CS is equal to square root of. A multiplied by A is a squared. And then we're multiplying E to the power of Cu by Ease the power of CV. Using the properties of exponents, we can show that this is E raises the power of CU plus CV. Now, what we can do is simply take square root of A2d because now we can show that on the right hand side we have square root of a squad. Multiplied by a square root of. He raised the power of CU plus CV. Square root of a squad if A is greater than 0 is simply a, so we can now show that a multiplied bye. Race the power of Cs is equal to a square root of. He raised the power of CU plus CV. We can now divide both sides by a. And what we can do from here is simply. Show that e to the power of cs is equal to square root of. E Was the power of. CV plus CU, right? And now what we can do in addition to that is factor out C. So we get E raises the power of C and U + V. Now let's go ahead and square both sides. Let's recall that if we have E raise the power of CS and we square it, we get E raises the power of 2 Cs. In other words, we're multiplying our exponents. So we get E, raises the power of 2 C S is equal to E, raises the power of C and print C U + V. Because our bases are equal to each other, we can equate our exponents to show that 2 Cs must be equal to c impres U plus V. We can now divide both sides by C, which is not equal to 0, and solve for s. So S is going to be equal to our right hand side, which is U + V. And we're going to divide it by 2, the coefficient in front of S. Therefore, S is equal to U + V divided by 2, which corresponds to the answer choice d. Thank you for watching.

$x$	$f (x)$
$0$	$3$
$1$	$6$
$2$	$12$
$3$	$24$

Geometric means A quantity grows exponentially according to y(t) = y₀eᵏᵗ. What is the relationship among m, n, and p such that y(p) = √(y(m)y(n))?

Key Concepts

Exponential Growth Function

Geometric Mean

Properties of Exponents and Logarithms

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