Question

a human heart beats approximately 36,500,000 times in a year. find the closest estimate for this number that is a product of a 1 - digit number and an integer power of 10. 36,500,000 is \\(\square \times 10^{\square}\\)

Explanation:

Step1: Identify the number

The number is 36,500,000.

Step2: Convert to scientific notation

We need to write it as a 1 - digit number times a power of 10. Move the decimal point to get a 1 - digit number. For 36,500,000, moving the decimal 7 places to the left gives \(3.65\times10^{7}\), but we need the closest estimate with a 1 - digit number. The closest 1 - digit number approximation for 3.65 is 4? Wait, no, wait. Wait, the number is 36,500,000. Let's do scientific notation properly. \(36500000=3.65\times10^{7}\), but if we take the closest 1 - digit number (the coefficient should be a 1 - digit number, so we can round 3.65 to 4? No, wait, maybe I misread. Wait, the problem says "the closest estimate for this number that is a product of a 1 - digit number and an integer power of 10". So first, let's express 36,500,000 in scientific notation. The standard form is \(a\times10^{n}\) where \(1\leq a<10\). So \(36500000 = 3.65\times10^{7}\). Now, we need to approximate 3.65 to a 1 - digit number. The closest 1 - digit number to 3.65 is 4? Wait, no, 3.65 is closer to 4? Wait, 3.65 is 3.65, the distance from 3 is 0.65, from 4 is 0.35. So yes, closer to 4? Wait, but maybe the problem is to write it as a 1 - digit number (so \(a\) is a 1 - digit number, 1 - 9) times \(10^{n}\). Wait, 36,500,000. Let's count the number of zeros and the non - zero digits. 36,500,000 has 7 digits after 3? Wait, 3 (1), 6 (2), 5 (3), 0 (4), 0 (5), 0 (6), 0 (7), 0 (8)? Wait, no, 36,500,000 is 3 followed by 6,5,0,0,0,0,0. So the number of digits is 8. So when we write in scientific notation, we move the decimal 7 places to the left: \(3.65\times10^{7}\). Now, to get a 1 - digit number (so \(a\) is a single digit), we can round 3.65 to 4, so \(4\times10^{7}\), or maybe the problem expects us to use 3.65 rounded to 4? Wait, no, maybe I made a mistake. Wait, 36,500,000. Let's see, \(36500000 = 3.65\times10^{7}\). If we take the coefficient as a 1 - digit number, the closest 1 - digit number to 3.65 is 4? Wait, but 3.65 is closer to 4 than to 3? The difference between 3.65 and 3 is 0.65, between 3.65 and 4 is 0.35. So yes, closer to 4. But wait, maybe the problem is not about rounding the coefficient, but about the power of 10. Wait, no, the number is 36,500,000. Let's do it step by step.

Wait, maybe I misread the number. The number is 36,500,000? Wait, the image shows "36,500,000". So let's write it as \(a\times10^{n}\), \(1\leq a<10\). So \(36500000=3.65\times10^{7}\). Now, we need to approximate \(3.65\) to a 1 - digit number. So \(a\) should be a 1 - digit number (1 - 9). So the closest 1 - digit number to 3.65 is 4? So \(4\times10^{7}\). But wait, maybe the problem is simpler. Let's count the number of zeros. 36,500,000. Let's write it as 3.65×10⁷. If we take the 1 - digit number as 4, then 4×10⁷. But maybe the intended answer is 4×10⁷? Wait, no, wait, maybe I made a mistake in the exponent. Let's count the number of digits after the first non - zero digit. 36,500,000: the first non - zero digit is 3. The number of digits from 3 to the end is 7 (6,5,0,0,0,0,0). So 10⁷. And the coefficient is 3.65, which is close to 4. So the closest estimate with a 1 - digit number is 4×10⁷. But wait, maybe the problem is to write it as a 1 - digit number (so 3, 4, etc.) times 10^n. Let's check: 3×10⁷ = 30,000,000; 4×10⁷ = 40,000,000. The original number is 36,500,000. The distance from 30,000,000 is 6,500,000; from 40,000,000 is 3,500,000. So 4×10⁷ is closer. So the 1 - digit number is 4 and the power of 10 is 7. Wait, but wait, maybe the problem is not about rounding the coefficient, but just expressing it as a 1 - digit number (so the coefficient is a 1 - digit number, so we can have 3.65, but 3.65 is not a 1 - digit number. Wait, the problem says "a product of a 1 - digit number and an integer power of 10". So the first factor must be a 1 - digit number (1 - 9), and the second factor is a power of 10. So we need to find the closest such number to 36,500,000. So let's list possible candidates:

- \(3\times10^{7}=30,000,000\)
- \(4\times10^{7}=40,000,000\)

Calculate the absolute differences:

\(\vert36500000 - 30000000\vert=6500000\)

\(\vert36500000 - 40000000\vert = 3500000\)

Since 3500000 < 6500000, \(4\times10^{7}\) is closer. So the 1 - digit number is 4 and the power of 10 is 7.

Wait, but maybe I made a mistake in the exponent. Let's check the number of digits. 36,500,000: how many times do we multiply by 10 to get from 3.65 to 36,500,000? \(3.65\times10^{7}=36500000\). So the exponent is 7. And the 1 - digit approximation of 3.65 is 4. So the closest estimate is \(4\times10^{7}\).

Answer:

\(4\times10^{7}\) (so the first box is 4 and the second box is 7)