Free Math Worksheets

- Convert both measurements to meters first. - Divide the larger thickness by the smaller thickness. - Subtract exponents when dividing powers of \(10\).

1. Convert the coating thickness: \(150\,\text{nm}=150\times10^{-9}\,\text{m}=1.5\times10^{-7}\,\text{m}\). 2. Convert the hair thickness: \(0.06\,\text{mm}=0.06\times10^{-3}\,\text{m}=6\times10^{-5}\,\text{m}\). 3. Divide the thicknesses: \(\frac{6\times10^{-5}}{1.5\times10^{-7}}=\frac{6}{1.5}\times10^{-5-(-7)}=4\times10^2=400\).

Coating: \(1.5\times10^{-7}\,\text{m}\). Hair: \(6\times10^{-5}\,\text{m}\). Approximately \(400\) layers.

- Convert both distances to kilometers. - Multiply the light-year conversion by \(12\). - Compare the coefficients when the powers of ten are the same.

1. Using the given approximation, estimate Star A's distance: \(12\cdot(9.5\times 10^{12})=114\times 10^{12}=1.14\times 10^{14}\,\text{km}\). Thus, Star A is approximately \(1.14\times 10^{14}\,\text{km}\) away. 2. Compare the distances: \(1.14\times 10^{14}>1.0\times 10^{14}\). 3. Therefore, Star A is farther from Earth.

Star A is farther away because its distance is approximately \(1.14\times 10^{14}\,\text{km}\), which is greater than \(1.0\times 10^{14}\,\text{km}\).

- Convert each actual measurement to the unit you want for the model. - Divide every actual length by \(10^9\). - Use exponent rules to simplify the powers of ten.

1. Using the approximate diameter of the Sun, \(1.4\times 10^6\,\text{km}=1.4\times 10^9\,\text{m}\). Dividing the rounded value by \(10^9\) gives a model diameter of approximately \(1.4\,\text{m}\). 2. Using the approximate diameter of Earth, \(1.28\times 10^4\,\text{km}=1.28\times 10^{10}\,\text{mm}\). Dividing the rounded value by \(10^9\) gives a model diameter of approximately \(12.8\,\text{mm}\). 3. Using the approximate Earth-Sun distance, \(1.5\times 10^8\,\text{km}=1.5\times 10^{11}\,\text{m}\). Dividing the rounded value by \(10^9\) gives a model distance of approximately \(150\,\text{m}\).

a) Approximately \(1.4\,\text{m}\) b) Approximately \(12.8\,\text{mm}\) c) Approximately \(150\,\text{m}\)

- Multiply the light-year value by \(4.2\). - Divide the actual distance by the kilometers represented by one model centimeter. - Convert centimeters to meters and kilometers.

1. Using the rounded light-year value, the estimated actual distance is \(4.2\cdot(1.0\times 10^{13})\approx 4.2\times 10^{13}\,\text{km}\), or approximately \(42{,}000{,}000{,}000{,}000\,\text{km}\). 2. The model length is approximately \((4.2\times 10^{13})\div(1.0\times 10^8)=4.2\times 10^5\,\text{cm}\). 3. Convert the model length: \(4.2\times 10^5\,\text{cm}=4200\,\text{m}=4.2\,\text{km}\).

a) Approximately \(42{,}000{,}000{,}000{,}000\,\text{km}\) b) Approximately \(4.2\,\text{km}\)