Studdy AI Logo
Math  /  Geometry

QuestionSports American football fields measure 100 yards long between the end zones, and are 531353 \frac{1}{3} yards wide. Is the length of the diagonal across this field more or less than 120 yards? Explain. \qquad

Studdy Solution

STEP 1

1. The football field is a rectangle.
2. The length of the field is 100 yards.
3. The width of the field is 5313 53 \frac{1}{3} yards.
4. We need to calculate the length of the diagonal and compare it to 120 yards.

STEP 2

1. Convert the width to an improper fraction or decimal.
2. Use the Pythagorean Theorem to find the diagonal.
3. Compare the diagonal length to 120 yards.

STEP 3

Convert the width of the field, 5313 53 \frac{1}{3} yards, to an improper fraction or decimal for easier calculation.
5313=53+13=1593+13=1603 53 \frac{1}{3} = 53 + \frac{1}{3} = \frac{159}{3} + \frac{1}{3} = \frac{160}{3}
Alternatively, as a decimal:
5313=53.3333… 53 \frac{1}{3} = 53.3333\ldots

STEP 4

Use the Pythagorean Theorem to find the diagonal of the rectangle. The formula for the diagonal d d is:
d=(length)2+(width)2 d = \sqrt{(\text{length})^2 + (\text{width})^2}
Substitute the values:
d=(100)2+(1603)2 d = \sqrt{(100)^2 + \left(\frac{160}{3}\right)^2}

STEP 5

Calculate the squares of the length and width:
(100)2=10000 (100)^2 = 10000
(1603)2=16029=256009 \left(\frac{160}{3}\right)^2 = \frac{160^2}{9} = \frac{25600}{9}

STEP 6

Add the squared values:
d=10000+256009 d = \sqrt{10000 + \frac{25600}{9}}
To add these, convert 10000 to a fraction with a denominator of 9:
10000=900009 10000 = \frac{90000}{9}
d=900009+256009 d = \sqrt{\frac{90000}{9} + \frac{25600}{9}}
d=1156009 d = \sqrt{\frac{115600}{9}}

STEP 7

Simplify the square root:
d=1156003 d = \frac{\sqrt{115600}}{3}
Calculate 115600 \sqrt{115600} :
115600=340 \sqrt{115600} = 340
Thus, the diagonal is:
d=3403≈113.33 d = \frac{340}{3} \approx 113.33

STEP 8

Compare the calculated diagonal length to 120 yards:
Since 113.33<120 113.33 < 120 , the diagonal is less than 120 yards.
The length of the diagonal across the field is approximately 113.33 yards, which is less than 120 yards.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord