Question

find the distance between the points t(13, 1.6) and v(5.4, 3.7). the exact distance between the two points is \boxed{}.

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). For points \( T(13, 1.6) \) and \( V(5.4, 3.7) \), \( x_1 = 13 \), \( y_1 = 1.6 \), \( x_2 = 5.4 \), \( y_2 = 3.7 \).

Step2: Calculate \( x_2 - x_1 \) and \( y_2 - y_1 \)

\( x_2 - x_1 = 5.4 - 13 = -7.6 \)
\( y_2 - y_1 = 3.7 - 1.6 = 2.1 \)

Step3: Square the differences

\( (-7.6)^2 = 57.76 \)
\( (2.1)^2 = 4.41 \)

Step4: Sum the squares

\( 57.76 + 4.41 = 62.17 \)

Step5: Take the square root

\( d = \sqrt{62.17} \) (or we can keep it in the form before taking the square root if we want exact, but usually we compute the decimal or simplify the radical. Wait, 62.17 is 6217/100, so \( \sqrt{\frac{6217}{100}}=\frac{\sqrt{6217}}{10}\approx7.885 \)) Wait, let's recalculate the differences: \( x_2 - x_1 = 5.4 - 13 = -7.6 \), squared is \( (-7.6)^2 = 7.6\times7.6 = 57.76 \). \( y_2 - y_1 = 3.7 - 1.6 = 2.1 \), squared is \( 2.1\times2.1 = 4.41 \). Sum: 57.76 + 4.41 = 62.17. Then square root of 62.17. Let's compute that: \( \sqrt{62.17} \approx 7.885 \). But if we want the exact form, it's \( \sqrt{(5.4 - 13)^2 + (3.7 - 1.6)^2}=\sqrt{(-7.6)^2 + (2.1)^2}=\sqrt{57.76 + 4.41}=\sqrt{62.17} \). Alternatively, we can write 62.17 as 6217/100, so \( \sqrt{\frac{6217}{100}}=\frac{\sqrt{6217}}{10} \). But maybe we can compute the decimal. Let's check: 7.88^2 = (7 + 0.88)^2 = 49 + 12.32 + 0.7744 = 62.0944. 7.89^2 = 7.88^2 + 27.880.01 + 0.01^2 = 62.0944 + 0.1576 + 0.0001 = 62.2521. So 62.17 is between 7.88^2 and 7.89^2. Let's do linear approximation. Let f(x)=x^2, f'(x)=2x. At x=7.88, f(x)=62.0944, f'(x)=15.76. We need to find Δx such that 62.0944 + 15.76Δx = 62.17. Δx=(62.17 - 62.0944)/15.76=0.0756/15.76≈0.0048. So x≈7.88 + 0.0048≈7.8848, so approximately 7.88. But the exact distance is \( \sqrt{62.17} \) or approximately 7.88. Wait, maybe I made a mistake in calculation. Wait 13 - 5.4 is 7.6, so (x2 - x1) is -7.6, squared is same as 7.6 squared. 7.67.6: 77=49, 70.6=4.2, 0.67=4.2, 0.60.6=0.36, so (7 + 0.6)^2=49 + 8.4 + 0.36=57.76. Correct. 3.7 - 1.6=2.1, squared is 4.41. Sum is 57.76 + 4.41=62.17. So square root of 62.17. Let's check with calculator: √62.17 ≈7.884795... So approximately 7.88 or 7.89. But the exact form is √62.17. Alternatively, we can write it as √(6217/100)=√6217 / 10. Since 6217 is a prime? Let's check: 6217 divided by 13: 13478=6214, 6217-6214=3, not divisible. 7: 7888=6216, 6217-6216=1, not divisible. 17: 17365=6205, 6217-6205=12, not divisible. So 6217 is prime? Maybe. So the exact distance is √62.17 or √6217 / 10, and the approximate is about 7.88.

Answer:

\( \sqrt{62.17} \) (or approximately \( 7.88 \))