Analyze the equation for the term with the greatest number of decimal places in the given equation. The power of 10 is equal to the greatest number of decimal places of any term in the given equation.

Question

asked 2021-05-19

For digits before decimals point, multiply each digit with the positive powers of ten where power is equal to the position of digit counted from left to right starting from 0.

For digits after decimals point, multiply each digit with the negative powers of ten where power is equal to the position of digit counted from right to left starting from 1.

1) \(10^{0}=1\)

2) \(10^{1}=10\)

3) \(10^{2}=100\)

4) \(10^{3}=1000\)

5) \(10^{4}=10000\)

And so on...

6) \(10^{-1}=0.1\)

7) \(10^{-2}=0.01\)

8) \(10^{-3}=0.001\)

9) \(10^{-4}=0.0001\)

For digits after decimals point, multiply each digit with the negative powers of ten where power is equal to the position of digit counted from right to left starting from 1.

1) \(10^{0}=1\)

2) \(10^{1}=10\)

3) \(10^{2}=100\)

4) \(10^{3}=1000\)

5) \(10^{4}=10000\)

And so on...

6) \(10^{-1}=0.1\)

7) \(10^{-2}=0.01\)

8) \(10^{-3}=0.001\)

9) \(10^{-4}=0.0001\)

asked 2021-08-08

When multiplying decimals, how do you know where to place the decimal point? Think about this as you do parts (a) through (d) below.

a. Write two equations for multiplying 0.3 by 0.16. One equation should express the factors and product using decimals, and one using fractions.

b. In part (a), you multiplied tenths by hundredths to get thousandths. Do you always get thousandths if you multiply tenths by hundredths? Why or why not? What do you get if you multiply tenths by tenths? Hundredths by hundredths? Use several examples to justify your answers.

c. When a multiplication problem is written using decimals, there is a relationship between the number of decimal places in the parts (or factors) of the problem and the number of decimal places in the answer (or product). Describe this relationship.

d. Describe a shortcut for locating the decimal point in the answer to a problem involving decimal multiplication.

a. Write two equations for multiplying 0.3 by 0.16. One equation should express the factors and product using decimals, and one using fractions.

b. In part (a), you multiplied tenths by hundredths to get thousandths. Do you always get thousandths if you multiply tenths by hundredths? Why or why not? What do you get if you multiply tenths by tenths? Hundredths by hundredths? Use several examples to justify your answers.

c. When a multiplication problem is written using decimals, there is a relationship between the number of decimal places in the parts (or factors) of the problem and the number of decimal places in the answer (or product). Describe this relationship.

d. Describe a shortcut for locating the decimal point in the answer to a problem involving decimal multiplication.

asked 2021-08-02

Ben and Carl are remembering how you told them that a 0 at the end of a whole number makes it ten times greater. They remember that 420 is ten times bigger than 42, for example. They wonder if the same thing works with decimals. Ben argues that putting a 0 at the end of a decimal number makes it ten times greater. Carl says, "No. You've gotta sneak it in right after the decimal point and push everything to the right." How do you help them to see what's true and what's not?

asked 2021-05-09

The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus

\(\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}\)

where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.

Part A:

Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.

Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.

Part B:

A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).

Assume the following:

The top speed is limited by air drag.

The magnitude of the force of air drag at these speeds is proportional to the square of the speed.

By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?

Express the percent increase in top speed numerically to two significant figures.

\(\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}\)

where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.

Part A:

Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.

Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.

Part B:

A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).

Assume the following:

The top speed is limited by air drag.

The magnitude of the force of air drag at these speeds is proportional to the square of the speed.

By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?

Express the percent increase in top speed numerically to two significant figures.