Find the approximate value of f (5.001),
Question: Find the approximate value of $f(5.001)$, where $f(x)=x^{3}-7 x^{2}+15$. Solution: Let: $x=5$ $x+\Delta x=5.001$ $\Rightarrow \Delta x=0.001$ $f(x)=x^{3}-7 x^{2}+15$ $\Rightarrow y=f(x=3)=125-175+15=-35$ Now, $y=f(x)$ $\Rightarrow \frac{d y}{d x}=3 x^{2}-14 x$ $\therefore d y=\Delta y=\frac{d y}{d x} d x=\left(3 x^{2}-14 x\right) \times 0.001=(75-70) \times 0.001=0.005$ $\therefore f(5.001)=y+\Delta y=-35+0.005=-34.995$...
Read More →Wild buffalo is an endangered species because.
Question: Wild buffalo is an endangered species because . (a) its population is diminishing (b) it has become extinct (c) it is found exclusively in a particular area (d) its poaching is strictly prohibited Solution: (a) Wild buffalo is an endangered species because its population is diminishing. Endangered species are the species which are facing the risk of extinction. Their numbers are decreasing ts such a low level that they might face extinction soon....
Read More →Find the sum of the series:
Question: Find the sum of the series: $\left(3 \times 1^{2}\right)+\left(5 \times 2^{2}\right)+\left(7 \times 3^{2}\right)+\ldots$ to $n$ terms Solution: In the given question we need to find the sum of the series. For that, first, we need to find the $\mathrm{n}^{\text {th }}$ term of the series so that we can use summation of the series with standard identities and get the required sum. The series given is $\left(3 \times 1^{2}\right)+\left(5 \times 2^{2}\right)+\left(7 \times 3^{2}\right)+\ld...
Read More →Find the approximate value of f (2.01),
Question: Find the approximate value of $f(2.01)$, where $f(x)=4 x^{2}+5 x+2$ Solution: Let: $x=2$ $x+\Delta x=2.01$ $\Rightarrow \Delta x=0.01$ $f(x)=4 x^{2}+5 x+2$ $\Rightarrow f(x=2)=16+10+2=28$ Now, $y=f(x)$ $\Rightarrow \frac{d y}{d x}=8 x+5$ $\therefore d y=\Delta y=\frac{d y}{d x} d x=(8 x+5) \times 0.01=(16+5) \times 0.01=0.21$ $\therefore f(2.01)=y+\Delta y=28.21$...
Read More →Find the sum of the series:
Question: Find the sum of the series: $\left(1 \times 2^{2}\right)+\left(3 \times 3^{2}\right)+\left(5 \times 4^{2}\right)+\ldots$ to $n$ terms Solution: In the given question we need to find the sum of the series. For that, first, we need to find the $\mathrm{n}^{\text {th }}$ term of the series so that we can use summation of the series with standard identities and get the required sum. The series given is $\left(1 \times 2^{2}\right)+\left(3 \times 3^{2}\right)+\left(5 \times 4^{2}\right)+\ld...
Read More →The money to be spent for the welfare of the employees of a firm
Question: The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of $x$ units of a product is given by $R(x)=3 x^{2}+36 x+5$, find the marginal revenue, when $x=5$, and write which value does the question indicate. Solution: Since, marginal revenue is the rate of change of total revenue with respect to the number of units sold, we have Marginal revenue ...
Read More →The money to be spent for the welfare of the employees of a firm
Question: The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of $x$ units of a product is given by $R(x)=3 x^{2}+36 x+5$, find the marginal revenue, when $x=5$, and write which value does the question indicate. Solution: Since, marginal revenue is the rate of change of total revenue with respect to the number of units sold, we have Marginal revenue ...
Read More →The total revenue received from the sale of x units of a product is given by
Question: The total revenue received from the sale of $x$ units of a product is given by $R(x)=13 x^{2}+26 x+15$. Find the marginal revenue when $x=7$. Solution: Since the marginal revenue is the rate of change of total revenue with respect to its output, Marginal Revenue $(\mathrm{MR})=\frac{d R}{d x}(x)=\frac{d}{d x}\left(13 x^{2}+26 x+15\right)=26 x+26$ When $x=7$ Marginal Revenue $(\mathrm{MR})=26(7)+26=182+26=$ Rs 208...
Read More →The total cost C (x) associated with the production of x units of an item is given by
Question: The total cost $C(x)$ associated with the production of $x$ units of an item is given by $C(x)=0.007 x^{3}-0.003 x^{2}+15 x+4000$. Find the marginal cost when 17 units are produced. Solution: Since the marginal cost is the rate of change of total cost with respect to its output, Marginal Cost (MC) $=\frac{d C}{d x}(x)=\frac{d}{d x}\left(0.007 x^{3}-0.003 x^{2}+15 x+4000\right)=0.021 x^{2}-0.006 x+15$ When $x=17$ Marginal Cost $(M C)==0.021(17)^{2}-0.006(17)+15=6.069-0.102+15=R s 20.967...
Read More →Find the sum of the series:
Question: Find the sum of the series: $\left(1 \times 2^{2}\right)+\left(2 \times 3^{2}\right)+\left(3 \times 4^{2}\right)+\ldots$ to $n$ terms Solution: In the given question we need to find the sum of the series. For that, first, we need to find the nth term of the series so that we can use summation of the series with standard identities and get the required sum. The series given is $\left(1 \times 2^{2}\right)+\left(2 \times 3^{2}\right)+\left(3 \times 4^{2}\right)+\ldots$ to $n$ terms. The ...
Read More →Find the rate of change of the volume of a ball with respect to its radius r.
Question: Find the rate of change of the volume of a ball with respect to its radiusr. How fast is the volume changing with respect to the radius when the radius is 2 cm? Solution: LetVbe the volume of the spherical ball. Then, $V=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d r}=4 \pi r^{2}$' Thus, the rate of change of the volume of the sphere is $4 \pi r^{2}$. When $r=2 \mathrm{~cm}$ $\left(\frac{d V}{d r}\right)_{r=2}=4 \pi(2)^{2}$ $=16 \pi \mathrm{cm}^{3} / \mathrm{cm}$...
Read More →Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm.
Question: Find the rate of change of the area of a circle with respect to its radiusrwhenr= 5 cm. Solution: LetAbe area of the circle. Then, $A=\pi r^{2}$ $\Rightarrow \frac{d A}{d r}=2 \pi r$ Hence, the rate of change of the area of the circle is2r2r.Whenr= 5 cm, $\left(\frac{d A}{d r}\right)_{r=5}=2 \pi(5)$ $=10 \pi \mathrm{cm}^{2} / \mathrm{cm}$...
Read More →Find the rate of change of the volume of a cone with respect to the radius of its base.
Question: Find the rate of change of the volume of a cone with respect to the radius of its base. Solution: LetVbe the volume of the cone. Then, $V=\frac{1}{3} \pi r^{2} h$ $\Rightarrow \frac{d V}{d r}=\frac{2}{3} \pi r h$...
Read More →Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm.
Question: Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm. Solution: LetAbe the area of the circular disc. Then, $A=\pi r^{2}$ $\Rightarrow \frac{d A}{d r}=2 \pi r$ LetCbe the circumference of the circular disc. Then, $C=2 \pi r$ $\Rightarrow \frac{d C}{d r}=2 \pi$ $\therefore \frac{d A}{d C}=\frac{d A / d r}{d C / d r}$ $\Rightarrow \frac{d A}{d C}=\frac{2 \pi r}{2 \pi}=r$ $\Rightarrow\left(\frac{d A}{d C}\right)_{r=3}=3 \mathrm{~...
Read More →Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm.
Question: Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm. Solution: LetVbe the volume of the sphere. Then, $V=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d r}=4 \pi r^{2}$ Let $S$ be the total surface area of sphere. Then, $S=4 \pi r^{2}$ $\Rightarrow \frac{d S}{d r}=8 \pi r$ $\therefore \frac{d V}{d S}=\frac{d V}{d r} / \frac{d S}{d r}$ $\Rightarrow \frac{d V}{d S}=\frac{4 \pi r^{2}}{8 \pi r}=\frac{r}{2}$ $\Rightarrow\left(\frac{d V...
Read More →Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm.
Question: Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm. Solution: LetVbe the volume of the sphere. Then, $V=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d r}=4 \pi r^{2}$ Let $S$ be the total surface area of sphere. Then, $S=4 \pi r^{2}$ $\Rightarrow \frac{d S}{d r}=8 \pi r$ $\therefore \frac{d V}{d S}=\frac{d V}{d r} / \frac{d S}{d r}$ $\Rightarrow \frac{d V}{d S}=\frac{4 \pi r^{2}}{8 \pi r}=\frac{r}{2}$ $\Rightarrow\left(\frac{d V...
Read More →Forest fire produces a lot of air pollution.
Question: Forest fire produces a lot of air pollution. Write in brief about the reasons of forest fires. Solution: Reasons of forest fires are : (i) At high temperature, sometimes dry grass catches fire which spreads throughout the forest. (ii) Camp fire may also be a reason. (iii) Due to the spark of lightning from the sky. (iv) The use of fires by villagers to ward off wild animals. (v) Fire lit intentionally by people living around forests for recreation. (vi) Fires started accidentally by ca...
Read More →Find the sum of the series:
Question: Find the sum of the series: $(3 \times 8)+(6 \times 11)+(9 \times 14)+\ldots$ to $n$ terms Solution: In the given question we need to find the sum of the series. For that, first, we need to find the nth term of the series so that we can use summation of the series with standard identities and get the required sum. The series given is (3 8) + (6 11) + (9 14) + to n terms. The series can be written as, [(3 x 1) x (3 x 1 + 5)), (3 x 2) x (3 x 2 + 5)) (3n x (3n + 5))]. So, $\mathrm{n}^{\te...
Read More →Although wood has a very high
Question: Although wood has a very high calorific value, we still discourage its use as a fuel. Explain. Solution: Burning of wood has several disadvantages. These are as follows: (i) Burning of wood produces a lot of smoke which causes respiratory diseases. (ii) The cutting down of trees to obtain as a wood fuel leads to deforestation which is very harmful to the environment. (iii) Trees provide us many-useful substances. To obtain fuel wood, when trees are cut down, then all useful substances ...
Read More →The calorific values of petrol
Question: The calorific values of petrol and CNG are 45000 kJ/kg and 50000 kJ/kg respectively. If you have vehicle which can run on petrol as well as CNG, which fuel will you prefer and why? Solution: We will prefer CNG (Compressed Natural Gas) because the calorific value of CNG is higher than that of petrol. CNG will produce large amount of heat energy than petrol. At the same lime, it produces the least air pollutants. CNG will be more economical....
Read More →Find the rate of change of the volume of a sphere with respect to its diameter.
Question: Find the rate of change of the volume of a sphere with respect to its diameter. Solution: LetVandrbe the volume and diameter of the sphere, respectively. Then, $V=\frac{4}{3} \pi(\text { radius })^{3}$ $\Rightarrow V=\frac{4}{3} \pi\left(\frac{r}{2}\right)^{3}=\frac{1}{6} \pi r^{3}$ $\Rightarrow \frac{d V}{d r}=\frac{1}{2} \pi r^{2}$...
Read More →Find the rate of change of the volume of a sphere with respect to its diameter.
Question: Find the rate of change of the volume of a sphere with respect to its diameter. Solution: LetVandrbe the volume and diameter of the sphere, respectively. Then, $V=\frac{4}{3} \pi(\text { radius })^{3}$ $\Rightarrow V=\frac{4}{3} \pi\left(\frac{r}{2}\right)^{3}=\frac{1}{6} \pi r^{3}$ $\Rightarrow \frac{d V}{d r}=\frac{1}{2} \pi r^{2}$...
Read More →Give two examples each for a solid,
Question: Give two examples each for a solid, liquid and gaseous fuel along with some important uses. Solution: Solid fuels Examples are wood and coal. These are used to cook food in homes. Coal is also used in industries. Liquid fuels Examples are kerosene and petrol. Kerosene is used in stoves to cook food and in lamps and petrol is used as a fuel in automobiles. Gaseous fuels Examples are natural gas and petroleum gas. These are used in . industries, CNG is used to run automobiles....
Read More →Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies.
Question: Find the rate of change of the total surface area of a cylinder of radiusrand heighth, when the radius varies. Solution: LetTbe the total surface area of a cylinder. Then, $T=2 \pi r(r+h)$ Since the radius varies, we differentiate the total surface area w.r.t. radiusr. Now, $\frac{d T}{d r}=\frac{d}{d r}[2 \pi r(r+h)]$ $\Rightarrow \frac{d T}{d r}=\frac{d}{d r}\left(2 \pi r^{2}\right)+\frac{d}{d r}(2 \pi r h)$ $\Rightarrow \frac{d T}{d r}=4 \pi r+2 \pi h$ $\Rightarrow \frac{d T}{d r}=2...
Read More →What are the three essential requirements
Question: What are the three essential requirements to produce fire? How fire extinguisher is useful for controlling the fire? Solution: Three essential requirements to produce fire are as follows: (i) Fuel (combustible substance) (ii) Air (or oxygen) and (iii) Heat to acquire the ignition temperature. The job of fire extinguisher is to cut off the supply of air or to bring down the temperature of fuel or both. The most common fire extinguisher is water. But water works only when things like woo...
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