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What is the largest n for which one can solve within 10 seconds a problem using an algorithm that requires f(n) bit operations, where each bit operation is carried out in 10-10 seconds, with these functions of n? a. log₂ n b. √n c. n7 d. 10n³

What is the largest n for which one can solve within 10 seconds a problem using an algorithm that requires f(n) bit operations, where each bit operation is carried out in 10-10 seconds, with these functions of n? a. log₂ n b. √n c. n7 d. 10n³

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 41E
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### Problem Complexity and Time Constraints What is the largest \( n \) for which one can solve within 10 seconds a problem using an algorithm that requires \( f(n) \) bit operations, where each bit operation is carried out in \( 10^{-10} \) seconds, with these functions of \( n \)? #### Functions of \( n \): **a.** \( \log_2 n \) **b.** \( \sqrt{n} \) **c.** \( n^7 \) **d.** \( 10n^3 \) --- To determine the largest \( n \), set up the inequality based on the time constraint: \[ f(n) \times 10^{-10} \leq 10 \] Solving for \( n \) will give the maximum size of \( n \) solvable in 10 seconds for each given function. For each function \( f(n) \), follow these steps: 1. \( \log_2 n \): \[ \log_2 n \times 10^{-10} \leq 10 \\ \log_2 n \leq 10^{11} \\ n \leq 2^{10^{11}} \] 2. \( \sqrt{n} \): \[ \sqrt{n} \times 10^{-10} \leq 10 \\ \sqrt{n} \leq 10^{11} \\ n \leq 10^{22} \] 3. \( n^7 \): \[ n^7 \times 10^{-10} \leq 10 \\ n^7 \leq 10^{11} \\ n \leq (10^{11})^{1/7} \\ n \leq 10^{11/7} \] 4. \( 10n^3 \): \[ 10n^3 \times 10^{-10} \leq 10 \\ n^3 \leq 10^{10} \\ n \leq 10^{10/3} \\ n \leq 10^{3.33} \] By calculating these, one can find the largest \( n \) that can be solved within the given constraints for each function
Transcribed Image Text:### Problem Complexity and Time Constraints What is the largest \( n \) for which one can solve within 10 seconds a problem using an algorithm that requires \( f(n) \) bit operations, where each bit operation is carried out in \( 10^{-10} \) seconds, with these functions of \( n \)? #### Functions of \( n \): **a.** \( \log_2 n \) **b.** \( \sqrt{n} \) **c.** \( n^7 \) **d.** \( 10n^3 \) --- To determine the largest \( n \), set up the inequality based on the time constraint: \[ f(n) \times 10^{-10} \leq 10 \] Solving for \( n \) will give the maximum size of \( n \) solvable in 10 seconds for each given function. For each function \( f(n) \), follow these steps: 1. \( \log_2 n \): \[ \log_2 n \times 10^{-10} \leq 10 \\ \log_2 n \leq 10^{11} \\ n \leq 2^{10^{11}} \] 2. \( \sqrt{n} \): \[ \sqrt{n} \times 10^{-10} \leq 10 \\ \sqrt{n} \leq 10^{11} \\ n \leq 10^{22} \] 3. \( n^7 \): \[ n^7 \times 10^{-10} \leq 10 \\ n^7 \leq 10^{11} \\ n \leq (10^{11})^{1/7} \\ n \leq 10^{11/7} \] 4. \( 10n^3 \): \[ 10n^3 \times 10^{-10} \leq 10 \\ n^3 \leq 10^{10} \\ n \leq 10^{10/3} \\ n \leq 10^{3.33} \] By calculating these, one can find the largest \( n \) that can be solved within the given constraints for each function
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