A company is designing a rectangular computer chip that is 8.14 × 10−7 meters long and 3.6 × 10−9 meters wide.
Part A: Determine the area of the computer chip. Show every step of your work.
Part B: Write the area in scientific notation with the correct number of significant digits, and label with the appropriate units. (3 points)
Part C: Determine the perimeter of the computer chip. Show every step of your work.
Part D: Write the perimeter in scientific notation with the correct number of significant digits, and label with the appropriate units.
Part A:
To find the area of the computer chip, we multiply the length and width together.
Area = length × width
Area = (8.14 × 10^-7 m) × (3.6 × 10^-9 m)
Area = 2.9324 × 10^-15 m^2
Part B:
The area, 2.9324 × 10^-15 m^2, has the correct number of significant digits since all the given numbers have significant digits. However, we need to rewrite it in scientific notation with the correct number of significant digits.
2.9324 × 10^-15 m^2 can be rewritten as 2.93 × 10^-15 m^2 (rounded to two significant digits).
Part C:
To find the perimeter of the computer chip, we add up all four sides.
Perimeter = 2(length + width)
Perimeter = 2(8.14 × 10^-7 m + 3.6 × 10^-9 m)
Perimeter = 2(8.1436 × 10^-7 m)
Perimeter = 1.62872 × 10^-6 m
Part D:
The perimeter, 1.62872 × 10^-6 m, has too many significant digits. We need to rewrite it in scientific notation with the correct number of significant digits.
1.62872 × 10^-6 m can be rewritten as 1.63 × 10^-6 m (rounded to two significant digits).
Part A:
To determine the area of the computer chip, we can use the formula for the area of a rectangle:
Area = length × width
Given that the length of the computer chip is 8.14 × 10^(-7) meters and the width is 3.6 × 10^(-9) meters, we can substitute these values into the formula:
Area = (8.14 × 10^(-7) m) × (3.6 × 10^(-9) m)
To multiply these numbers, we can perform the multiplication of the coefficients and add the exponents:
Area = (8.14 × 3.6) × (10 ^ (-7 + -9)) m^2
Area = 29.304 × 10^(-16) m^2
Part B:
To write the area in scientific notation with the correct number of significant digits and label with the appropriate units, we need to round the coefficient to the correct number of significant digits.
Since the greatest number of significant digits in the given values is 3, we should round our result to 3 significant digits:
Area = 29.3 × 10^(-16) m^2
Part C:
To determine the perimeter of the computer chip, we can use the formula for the perimeter of a rectangle:
Perimeter = 2 × (length + width)
Given that the length of the computer chip is 8.14 × 10^(-7) meters and the width is 3.6 × 10^(-9) meters, we can substitute these values into the formula:
Perimeter = 2 × (8.14 × 10^(-7) m + 3.6 × 10^(-9) m)
To add these numbers, we need to have them in the same exponent form:
Perimeter = 2 × (8.14 × 10^(-7) m + 0.036 × 10^(-7) m)
Perimeter = 2 × (8.14 + 0.036) × 10^(-7) m
Perimeter = 16.352 × 10^(-7) m
Part D:
To write the perimeter in scientific notation with the correct number of significant digits and label with the appropriate units, we need to round the coefficient to the correct number of significant digits.
Since the greatest number of significant digits in the given values is 3, we should round our result to 3 significant digits:
Perimeter = 16.4 × 10^(-7) m
Part A: To determine the area of the computer chip, you need to multiply the length and width.
Given:
Length = 8.14 × 10^(-7) meters
Width = 3.6 × 10^(-9) meters
Area = Length × Width
Substituting the given values:
Area = (8.14 × 10^(-7)) × (3.6 × 10^(-9))
To multiply numbers in scientific notation, multiply the coefficients and add the exponents:
Area = (8.14 × 3.6) × (10^(-7) × 10^(-9))
Calculating the coefficient: 8.14 × 3.6 = 29.304
Simplifying the exponents: 10^(-7) × 10^(-9) = 10^(-7-9) = 10^(-16)
Area = 29.304 × 10^(-16) square meters
Part B: To write the area in scientific notation with the correct number of significant digits, count the significant digits in the coefficient (29.304) and adjust the exponent accordingly. The exponent should be such that there is only one digit before the decimal point.
Counting the significant digits: 29.304 has five significant digits.
Writing in scientific notation with one significant digit before the decimal point:
Area = 2.9304 × 10^(-15) square meters
Part C: To determine the perimeter of the computer chip, you need to calculate the sum of all four sides.
Given:
Length = 8.14 × 10^(-7) meters
Width = 3.6 × 10^(-9) meters
Perimeter = 2 × (Length + Width)
Substituting the given values:
Perimeter = 2 × (8.14 × 10^(-7) + 3.6 × 10^(-9))
To add numbers in scientific notation, the coefficients must be similar. Let's rewrite the second term with an exponent of -7:
Perimeter = 2 × (8.14 × 10^(-7) + 3.6 × 10^(-7))
Calculating the coefficients: 8.14 + 3.6 = 11.74
Perimeter = 2 × 11.74 × 10^(-7) meters
Part D: To write the perimeter in scientific notation with the correct number of significant digits, count the significant digits in the coefficient (11.74) and adjust the exponent accordingly. The exponent should be such that there is only one digit before the decimal point.
Counting the significant digits: 11.74 has four significant digits.
Writing in scientific notation with one significant digit before the decimal point:
Perimeter = 1.2 × 10^(-6) meters