An airplane flight from Houston, Texas to Rome, Italy is 5,686 miles. Estimate the distance between the two cities as the product of a single digit number times a power of 10.(1 point)

Question

Responses

6×104
6 times 10 to the 4th power

5×104
5 times 10 to the 4th power

5×103
5 times 10 cubed

6×103
6 times 10 cubed

Maria wanted to bring back 5 million grains of sand from her trip to the beach. Each grain of sand weighs approximately 6.7×10−7
kg. How heavy is the bag of sand that Maria brings home?(1 point)
Responses

Multiply 5,000,000(6.7×10−7)=(5×106)⋅(6.7×10−7)=(5⋅6.7)(106+(−7))=33.5×10−1=3.35
kg
Multiply 5 comma 000 comma 000 times open paren 6 point 7 times 10 to the negative 7 power close paren is equal to open paren 5 times 10 to the 6th power close paren times open paren 6 point 7 times 10 to the negative 7 power close paren is equal to open paren 5 times 6 point 7 close paren times open paren 10 raised to the 6 plus negative 7 power close paren is equal to 33 point 5 times 10 inverse is equal to 3 point 3 5 kg

Multiply 5(6.7×10−7) = 33.5×10−7=3.35×10−6
kg
Multiply 5(6.7×10−7) = 33.5×10−7=3.35×10−6
kg

Divide 5,000,0006.7×10−7=5×1066.7×10−7=(56.7)×106+(−7)=0.74×10−1=0.074
kg
Divide the fraction with numerator 5 comma 000 comma 000 and denominator 6 point 7 times 10 to the negative 7 power is equal to the fraction with numerator 5 times 10 to the 6th power and denominator 6 point 7 times 10 to the negative 7 power is equal to open paren 5 over 6 point 7 close paren times 10 raised to the 6 plus negative 7 power is equal to 0 point 7 4 times 10 inverse is equal to 0 point 0 7 4 kg

Multiply 5,000,000(6.7×10−7)=(5×106)⋅(6.7×10−7)=(5⋅6.7)(106+7)=33.5×1013=3.35×1014
kg
Multiply 5 comma 000 comma 000 times open paren 6 point 7 times 10 to the negative 7 power close paren is equal to open paren 5 times 10 to the 6th power close paren times open paren 6 point 7 times 10 to the negative 7 power close paren is equal to open paren 5 times 6 point 7 close paren times open paren 10 raised to the 6 plus 7 power close paren is equal to 33 point 5 times 10 to the 13th power is equal to 3 point 3 5 times 10 to the 14th power kg

The Thai Pepper has a hotness rating of 7.5×104
on the Scoville scale.

The Red Savina Habanero has a hotness rating of 4.6×105
on the Scoville scale.

How many times hotter is the Red Savina Habanero than the Thai Pepper?

(1 point)
Responses

Divide (4.6×105)(7.5×104)=4.67.5×105−4=0.613×101=6.13
times hotter
Divide (4.6×105)(7.5×104)=4.67.5×105−4=0.613×101=6.13
times hotter

Multiply (7.5×104)⋅(4.6×105)=(7.5×4.6)⋅(104⋅5)=34.5×1020=3.45×1021
times hotter
Multiply open paren 7 point 5 times 10 to the 4th power close paren times open paren 4 point 6 times 10 to the 5th power close paren is equal to open paren 7 point 5 times 4 point 6 close paren times open paren 10 raised to the 4 times 5 power close paren is equal to 34 point 5 times 10 to the 20th power is equal to 3 point 4 5 times 10 to the 21th power times hotter

Divide (7.5×104)(4.6×105)=7.54.6×104−5=1.63×101=16.3
times hotter
Divide (7.5×104)(4.6×105)=7.54.6×104−5=1.63×101=16.3
times hotter

Multiply (7.5×104)⋅(4.6×105)=(7.5×4.6)⋅(104+5)=34.5×109=3.45×1010
times hotter

Mercury is 5.8×107
km from the sun. Neptune is 4.5×1010
km from the sun. How much further is Neptune from the sun than Mercury?(1 point)
Responses

Subtract the distances by getting the same power of 10: (4.5×1010)−(5.8×107) =(4.5×1010)−(0.0058×1010)=4.49×1010
km
Subtract the distances by getting the same power of 10: (4.5×1010)−(5.8×107) =(4.5×1010)−(0.0058×1010)=4.49×1010
km

Add the distances together: (5.8×107)+(4.5×1010) = 10.3×1017=1.03×1018
km
Add the distances together: (5.8×107)+(4.5×1010) = 10.3×1017=1.03×1018
km

Multiply the distances: (5.8×107)⋅(4.5×1010) = (5.8×4.5)⋅(107+10)=26.1×1017=2.61×1018
km
Multiply the distances: (5.8×107)⋅(4.5×1010) = (5.8×4.5)⋅(107+10)=26.1×1017=2.61×1018
km

Divide the distances: 4.5×10105.8×107=0.78×103 =7.8×102
km

When adding and subtracting numbers in Scientific Notation, the powers of the base 10 must be the same exponent so that you have "like terms".

When multiplying and dividing numbers in Scientific Notation, you do not have to have the same power of the base 10. You can simply follow the exponent rules for multiplying and dividing with the like base 10.

(1 point)
Responses

True, coefficients follow the operations and exponents follow exponent rules when you are multiplying or dividing.
True, coefficients follow the operations and exponents follow exponent rules when you are multiplying or dividing.

False, you must always have the same exponent on the base 10 for all operations in Scientific Notation.
False, you must always have the same exponent on the base 10 for all operations in Scientific Notation.

True, when multiplying and dividing you can just multiply the coefficients and multiply the exponents.
True, when multiplying and dividing you can just multiply the coefficients and multiply the exponents.

False, you cannot multiply or divide numbers in Scientific notation without having the same coefficients.

A cab company charges $12 per mile for a lift to the airport.

What change would the company make to their charges to make this a non proportional situation?

(1 point)
Responses

Charge $15 per mile instead of $12
Charge $15 per mile instead of $12

Charge $4 per mile instead of $12
Charge $4 per mile instead of $12

No changes are needed
No changes are needed

Charge a flat rate of $20 and then $12 per mile
Charge a flat rate of $20 and then $12 per mile

Answer 1

Let's analyze each question and provide the correct answers along with explanations:

1. Estimate the distance between Houston and Rome as a product of a single digit number times a power of 10.

   • The distance is 5,686 miles, which can be estimated to 6,000 miles, or \(6 \times 10^3\).
   • Correct Response: 6×10³

2. How heavy is the bag of sand Maria brings home?

   • To find the weight, multiply the number of grains (5,000,000) by the weight of each grain (6.7×10⁻⁷ kg): \[ 5,000,000 \times (6.7 \times 10^{-7}) = (5 \times 10^6) \times (6.7 \times 10^{-7}) = 33.5 \times 10^{-1} = 3.35 \text{ kg} \]
   • Correct Response: Multiply 5,000,000(6.7×10−7)=(5×10^6)⋅(6.7×10−7)=(5⋅6.7)(10^{6+(-7)})=33.5×10^{-1}=3.35 kg

3. How many times hotter is the Red Savina Habanero than the Thai Pepper?

   • To find out how many times hotter, divide the hotness ratings: \[ \frac{4.6 \times 10^5}{7.5 \times 10^4} = \frac{4.6}{7.5} \times 10^{5-4} = 0.613 \times 10^1 = 6.13 \]
   • Correct Response: Divide (4.6×10^5)(7.5×10^4)=4.67.5×10^{5−4}=0.613×10^1=6.13 times hotter

4. How much further is Neptune from the sun than Mercury?

   • We need to subtract the distances: \[ 4.5 \times 10^{10} - 5.8 \times 10^{7} = 4.5 \times 10^{10} - 0.0058 \times 10^{10} = 4.49 \times 10^{10} \text{ km} \]
   • Correct Response: Subtract the distances by getting the same power of 10: (4.5×10^{10})−(5.8×10^{7}) =(4.5×10^{10})−(0.0058×10^{10})=4.49×10^{10} km

5. True or False: When adding and subtracting numbers in Scientific Notation, the powers of the base 10 must be the same exponent.

   • Correct Response: True, coefficients follow the operations and exponents follow exponent rules when you are multiplying or dividing.

6. What change would make the cab company's charges a non-proportional situation?

   • A non-proportional situation occurs when there is a flat fee. Charging a flat rate of $20 and then $12 per mile breaks the direct proportion.
   • Correct Response: Charge a flat rate of $20 and then $12 per mile.

These answers reflect the appropriate applications of mathematical concepts using scientific notation, which is a common expectation in science and math at higher levels of education.