Question

1. the area of kaitlyn’s square garden is 345 square feet. one side of the garden is next to a shed. she wants to put a fence around the other three sides of the garden. find three sets of approximations for the amount of fence it will take. then determine how much fence she should buy. (example 3) 10. in little league, the bases are squares with sides of 14 inches. the expression \\(\sqrt{(s^2 + s^2)}\\) represents the distance diagonally across a square of side length \\(s\\). estimate the diagonal distance across a base to the nearest inch. (example 4)

Explanation:

Problem 9

Step1: Find side length of square

The area of a square is \( A = s^2 \), where \( s \) is the side length. Given \( A = 345 \) square feet, so \( s=\sqrt{345} \). We know that \( 18^2 = 324 \) and \( 19^2 = 361 \), so \( \sqrt{345} \) is between 18 and 19.

Step2: Approximation 1: Use 18 as side length

If \( s \approx 18 \), then the length of the fence (three sides) is \( 3\times18 = 54 \) feet.

Step3: Approximation 2: Use 18.5 as side length

Calculate \( 18.5^2=342.25 \), which is close to 345. So \( s\approx18.5 \), then the fence length is \( 3\times18.5 = 55.5 \) feet.

Step4: Approximation 3: Use 18.6 as side length

Calculate \( 18.6^2 = 18.6\times18.6 = 345.96 \), which is very close to 345. So \( s\approx18.6 \), then the fence length is \( 3\times18.6 = 55.8 \) feet.

Step5: Determine the exact amount to buy

Since \( \sqrt{345}\approx18.57 \), then three sides: \( 3\times18.57\approx55.71 \). So she should buy approximately 56 feet (rounding up to be safe) or more precisely, around 55.7 feet. But usually, we round up to the next whole number, so 56 feet.

Step1: Substitute \( s = 14 \) into the formula

The formula for the diagonal is \( \sqrt{s^2 + s^2} \). Substitute \( s = 14 \), we get \( \sqrt{14^2+14^2}=\sqrt{196 + 196}=\sqrt{392} \).

Step2: Estimate \( \sqrt{392} \)

We know that \( 19^2 = 361 \) and \( 20^2 = 400 \). \( 392 \) is closer to \( 400 \) than to \( 361 \). Also, \( 19.8^2=(20 - 0.2)^2=400-8 + 0.04 = 392.04 \), which is very close to 392. So \( \sqrt{392}\approx19.8\approx20 \) when rounded to the nearest inch.

Answer:

Approximations: 54 feet (using \( s\approx18 \)), 55.5 feet (using \( s\approx18.5 \)), 55.8 feet (using \( s\approx18.6 \)). She should buy approximately 56 feet of fence.

Problem 10