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This is one of our many quizzes on Exponential vs linear growth. Click on this to access more such competitive quizzes on Calculus.

Exponential vs linear growth Quiz 2

mathematics

 Identify the statements which are true about the exponential function: $y = 400 + {5^x}$  $ \\ $ $(I)$ The y-intercept of this graph is $401$ $ \\ $ $(II)$ The base in this equation is $400$ $ \\ $ $(III)$ The x-intercept of this equation is $−1$

 Know the general form of exponential function and then check all the three statements.

Given, $y = 400 + {5^x}$ $ \\ $ When, $x = 0$, $y = 401$ $ \\ $ Hence, the statement $(I)$ is correct. $ \\ $ Base of the function is $5$, $ \\ $ Hence, the statement $(II)$ is incorrect. $ \\ $ Exponential functions have x-intercept only when they are in the form, $ \\ \Rightarrow y = {a^x} + b$  Hence, the statement $(III)$ is incorrect.

 The population of an insect in a forest has been decreasing by $7$ every $4$ years since $2010$. The population at the beginning of $2010$ was $1,500$. If $M$ represents the total population of the insect at time $t$ years after $2010$, which of the following equations most closely describes the total population of the insect at any given time?

 Solve by trial-and-error methods by eliminating options by just putting values of $t$. Understand the given case and put the values of $t$ and get the answer.

W.K.T, Total population is a percentage of previous year’s population, Hence, It is an exponential function. $ \\ $ Let us assume, $ \\ $ Total population $= 100$  $ \\ $ Next year, Total population $= 93$ $ \\ $ The variable of time would be $\dfrac{t}{4}$ as population is decreasing after every $4$ year. $ \\ $ Hence $\left( b \right)M\left( t \right) = 1500{\left( {0.93} \right)^{\frac{t}{4}}}$

 Which of the following equations most closely describes the below graph? $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a20aba8f-3293-49ea-aacf-7f8961d2c1792816987121177697262.png"/>

 Solve by trial-and-error methods by eliminating options by just putting values of $x$ and getting the corresponding value of $y$. From that we can compare points with graphs. 

When $x = 0$, then $y = 2$ $ \\ $ Solving all options we get option $(b)$ as correct $ \\  \Rightarrow y = 2\left( {{2^x}} \right) \\   \Rightarrow y = 2\left( {{2^0}} \right) \\   \Rightarrow y = 2 \\  $

 The following equation represents the growth of bacteria in a dish: $C\left( t \right) = 50\left( {{2^t}} \right)$. Which of the following best describes this function?

 Use  trial-and-error methods by eliminating options by just putting some values of $t$and observing the pattern.

 Solve this question by eliminating the options. $ \\ $ Option $(a)$ and Option $(b)$ can be eliminated, as the function is exponential. $ \\ $ $t$ is an exponent with base $2$ $ \\ $ When, $t = 2$ $ \\ C\left( 2 \right) = 50\left( {{2^2}} \right) \\   \Rightarrow 50\left( 2 \right)\left( 2 \right) \\ \Rightarrow 200 \\  $ When, $t = 3$ $ \Rightarrow C\left( 3 \right) = 50\left( {{2^3}} \right) \\   \Rightarrow 50\left( 2 \right)\left( 2 \right)(2) \\   \Rightarrow 400 \\  $ So, the relationship is exponential and doubles every hour.

 Suppose you put stones into a mug hanging from a rubber band. The distance d from the floor in centimetres gets decreased as the number n of stones is increased. The data is mentioned in the table given below. If this pattern continues, what is the minimum number of stones needed to make the mug reach the floor? $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a60d5162-570e-48f7-98f7-bce2c276b04a2204029049320584840.png"/>

 Observe the pattern and then form the general equation.

Observe the pattern and form a general equation for the pattern. $ \\ $ Each stone is increasing the distance and decreasing by $2.5cm$. $ \\ $ So the general equation would be, $ \\ $ Distance from floor $= 45 - 2.5n$ $ \\ $ Distance from floor $=0$  $ \\ \Rightarrow 0 = 45 - 2.5n \\ \Rightarrow 45 = 2.5n \\ \Rightarrow n = \dfrac{{45}}{{2.5}} \\ \Rightarrow n = 18$