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Practice each question, then open the answer or worked solution.
One Fermi is equal to 10–15 metre. The radius of a proton is 1.3 Fermis. Write the radius of a proton in metres in standard form.
Ans: The radius of a proton is 1.3 fermi.
One fermi is equal to 10-15 m.
So, the radius of the proton is 1.3 x 10-15 m.
Hence, standard form of radius of the proton is 1.3 x 10-15 m.
The paper clip below has the indicated length. What is the length in standard form?
Ans: Length of the paper clip = 0.05 m
In standard form, 0.05 m = 0.5 x 10-1 = 5.0 x 10-2 m
Hence, the length of the paper clip in standard form is 5.0 x 10-2 m
Use the properties of exponents to verify that each statement is true.
(a) $\dfrac{1}{4}\left( {{2^n}} \right) = {2^{n - 2}}$
(b) ${4^{n - 1}} = \dfrac{1}{4}{\left( 4 \right)^n}$
(c) 25(5n–2) = 5n
Ans: ${\text{LHS = }}\dfrac{1}{4}\left( {{2^n}} \right) = \dfrac{1}{{{2^2}}} \times {2^n} = {2^{n - 2}} = {\text{RHS}}$
Ans: ${\text{RHS = }}\dfrac{1}{4}{\left( 4 \right)^n} = \dfrac{1}{4} \times {4^n} = {4^{n - 1}} = {\text{LHS}}$
Ans: ${\text{LHS = 25}}\left( {{5^{n - 2}}} \right) = {5^2}{5^{n - 2}} = {5^{2 + n - 2}} = {5^n} = {\text{RHS}}$
Fill in the blanks
Ans: $144 \times {2^{ - 3}} = 144 \times \dfrac{1}{8} = 18$
$18 \times {2^{ - 1}} = 18 \times \dfrac{1}{12} = \dfrac{3}{2}$
$\dfrac{3}{2} \times {3^{ - 2}} = \dfrac{3}{2} \times \dfrac{1}{{{3^2}}} = \dfrac{1}{{2 \times 3}} = \dfrac{1}{6}$
There are 864,00 seconds in a day. How many days long is a second? Express your answer in scientific notation.
Ans: Total seconds in a day = 86400
So, a second is long as $\dfrac{1}{{86400}} = 0.000011574$
Scientific notation of 0.000011574= 1.1574 x 10-5days
The given table shows the crop production of a State in the year 2008 and 2009. Observe the table given below and answer the given questions.
Crop | 2008 Harvest (Hectare) | Increase/Decrease (Hectare) in 2009 |
Bajra | 1.4 × 103 | – 100 |
Jowar | 1.7 × 106 | – 440,000 |
Rice | 3.7 × 103 | – 100 |
Wheat | 5.1 × 105 | + 190,000 |
(a) For which crop(s) did the production decrease?
(b) Write the production of all the crops in 2009 in their standard form.
(c) Assuming the same decrease in rice production each year as in 2009, how many acres will be harvested in 2015? Write in standard form.
Ans: From the given table we can figure out that bajra, jowar and rice crop’s production decreased.
Ans: The production of all crops in 2009
Bajra= 1.4 x 103 – 0.1 x 103 = 1.3 x 103
Jowar= 1.7 x 106 – 44 x 104
= 1.7 x 106 – 0.44 x 106 = 1.26 x 106
Rice= 3.7 x 103 – 0.1 x 103 = 3.6 x 103
Wheat= 5.1 x 105 + 19 x 104
= 5.1 x 105 + 1.9 x 105 = 7 x 105
Ans: Incomplete information.
Stretching Machine Suppose you have a stretching machine which could stretch almost anything. For example, if you put a 5 metre stick into a (× 4) stretching machine (as shown below), you get a 20 metre stick. Now if you put a 10 cm carrot into a (× 4) machine, how long will it be when it comes out?
Ans: According to the question, if we put a 5 m stick into a (x 4) stretching machine, then the machine produces 20 m sticks.
Similarly, if we put 10 cm carrots into a (x 4) stretching machine, then the machine produces 10 x 4 = 40 cm sticks.
Two machines can be hooked together. When something is sent through this hook up, the output from the first machine becomes the input for the second.
(a) Which two machines hooked together do the same work a (× 102) machine does? Is there more than one arrangement of two machines that will work?
(b) Which stretching machine does the same work as two (× 2) machines hooked together?
Ans:
To get the same work in (x102) machine, hook (x 22) and (x 52) machine.
So, x 102 = x 100
Now, we get,
22 x 52 = 4 x 25 = 100
Ans:
The same work would be produced by a (x 4) single machine when two machines (x 2) and (x 2) are hooked together.
Repeater Machine Similarly, a repeater machine is a hypothetical machine which automatically enlarges items several times. For example, sending a piece of wire through a (× 24) machine is the same as putting it through a (× 2) machine four times. So, if you send a 3 cm piece of wire through a (× 24) machine, its length becomes 3 × 2 × 2 × 2 × 2 = 48 cm. It can also be written that a base (2) machine is being applied 4 times.
Ans: According to the question, if we put a 3 cm piece of wire through a (x 24) machine, its length becomes 3 x 2 x 2 x 2 x 2 = 48 cm.
Similarly, a 4 cm long strip becomes 4 x 2 x 2 x 2 x 2 = 64 cm.
For the following repeater machines, how many times the base machine is applied and how much the total stretch is?
(a)
(b)
(c)
Ans: In the first machine that is (a), (x1002) = 10000 stretch. Because it is twice the base machine.
Ans: In machine (b), (x 75) = 16807 stretch.
Since, it is fair times the base machine.
Ans: In machine (c), (x 57) = 78125 stretch.
Since, it is 7 times the base machine.