A country has a food deficit of 10%....

MathematicsLinear Differential EquationsJEE Advanced 2000Moderate

A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to $\frac{l n 10 - l n 9}{l n ( 1.04 ) - 0.03}$ .

Visualized Solution (Hindi)

Defining the Variables

Let $P_{0}$ be the initial population at $t = 0$ .
Let $a$ be the constant average food requirement per person.
Initial food required: $R (0) = a P_{0}$ .
Given a $10%$ deficit, initial food production: $F (0) = 0.9 a P_{0}$ .

Modeling Population Growth

Population growth is continuous at $3%$ per year.
$\frac{d P}{d t} = 0.03 P \Rightarrow P (t) = P_{0} e^{0.03 t}$ .
Total food required at time $t$ : $R (t) = a P_{0} e^{0.03 t}$ .

Modeling Food Production

Food production grows annually by $4%$ .
This is a discrete growth model: $F (t) = F (0) (1 + 0.04)^{t}$ .
Substituting $F (0)$ : $F (t) = 0.9 a P_{0} (1.04)^{t}$ .

The Self-Sufficiency Condition

Self-sufficiency occurs when $F (t) \geq R (t)$ .
Substitute the expressions: $0.9 a P_{0} (1.04)^{t} \geq a P_{0} e^{0.03 t}$ .
Cancel common terms $a P_{0}$ : $0.9 (1.04)^{t} \geq e^{0.03 t}$ .

Applying Logarithms

Take natural log on both sides: $ln (0.9) + t ln (1.04) \geq 0.03 t$ .
Rearrange terms: $t [ln (1.04) - 0.03] \geq - ln (0.9)$ .
Since $- ln (0.9) = ln (\frac{1}{0.9}) = ln (\frac{10}{9}) = ln 10 - ln 9$ .

Final Calculation and Conclusion

Isolate $t$ : $t \geq \frac{l n 10 - l n 9}{l n ( 1.04 ) - 0.03}$ .
Since $n$ is the number of years, $n$ is the smallest integer $\geq t$ .
Key Takeaway: Higher production growth rate eventually overcomes initial deficits despite population growth.

00:00 / 00:00

Conceptually Similar Problems

MathematicsMonotonicityJEE Advanced 2003Moderate

View in:English Hindi

If $P (1) = 0$ and $d P (x) / d x > P (x)$ for all $x \geq 1$ then prove that $P (x) > 0$ for all $x > 1$ .

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1984Easy

View in:English Hindi

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

MathematicsVariable Separable MethodJEE Main 2014Easy

View in:English Hindi

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ equals:

(A)

600 - 500 e^{t /2}

(B)

400 - 300 e^{- t /2}

(C)

400 - 300 e^{t /2}

(D)

300 - 200 e^{- t /2}

Answer:C

MathematicsTheory of IndicesJEE Advanced 1980Easy

View in:English Hindi

Given $n^{4} < 1 0^{n}$ for a fixed positive integer $n \geq 2$ , prove that $(n + 1)^{4} < 1 0^{n + 1}$ .

MathematicsNewton-Leibniz & Reduction FormulasJEE Advanced 1996Moderate

View in:English Hindi

Let $A_{n}$ be the area bounded by the curve $y = (tan x)^{n}$ and the lines $x = 0, y = 0$ and $x = \frac{π}{4}$ . Prove that for $n > 2, A_{n} + A_{n - 2} = \frac{1}{n - 1}$ and deduce $\frac{1}{2 n + 2} < A_{n} < \frac{1}{2 n - 2}$ .

MathematicsGeometric Progression (G.P.)JEE Advanced 2006Moderate

View in:English Hindi

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

Answer:6

MathematicsVariable Separable MethodJEE Main 2012Easy

View in:English Hindi

The population $p (t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{d p ( t )}{d t} = 0.5 p (t) - 450$ . If $p (0) = 850$ , then the time at which the population becomes zero is :

(A)

2 ln 18

(B)

ln 9

(C)

\frac{1}{2} ln 18

(D)

ln 18

Answer:A

MathematicsProperties of Binomial CoefficientsJEE Advanced 1989Moderate

View in:English Hindi

Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

MathematicsGeometric Progression (G.P.)JEE Advanced 2001Moderate

View in:English Hindi

Let $a, b, c$ be positive real numbers such that $b^{2} - 4 a c > 0$ and let $α_{1} = c$ . Prove by induction that $α_{n + 1} = \frac{a α _{n}^{2}}{b ^{2} - 2 a ( α _{1} + α _{2} + .... + α _{n} )}$ is well-defined and $α_{n + 1} < \frac{α _{n}}{2}$ for all $n = 1, 2, ...$ (Here, 'well-defined' means that the denominator in the expression for $α_{n + 1}$ is not zero.)

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2002Easy

View in:English Hindi

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

(A)

n (2 c)^{1/ n}

(B)

(n + 1) c^{1/ n}

(C)

2 n c^{1/ n}

(D)

(n + 1) (2 c)^{1/ n}

Answer:A

Visualized Solution (Hindi)

Defining the Variables

Modeling Population Growth

Modeling Food Production

The Self-Sufficiency Condition

Applying Logarithms

Final Calculation and Conclusion

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(1)=0 and dP(x)/dx>P(x) for all x≥1 then prove that P(x)>0 for all x>1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the population of rabbits surviving at time t be governed by the differential equation dtdp(t)​=21​p(t)−200. If p(0)=100, then p(t) equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given n4<10n for a fixed positive integer n≥2, prove that (n+1)4<10n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let An​ be the area bounded by the curve y=(tanx)n and the lines x=0,y=0 and x=4π​. Prove that for n>2,An​+An−2​=n−11​ and deduce 2n+21​<An​<2n−21​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=43​−(43​)2+(43​)3+⋯+(−1)n−1(43​)n and bn​=1−an​, then find the least natural number n0​ such that bn​>an​ for all n≥n0​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that C0​−22C1​+32C2​−....+(−1)n(n+1)2Cn​=0,n>2, where Cr​=nCr​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,…,an​ are positive real numbers whose product is a fixed number c, then the minimum value of a1​+a2​+⋯+an−1​+2an​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(1)=0 and dP(x)/dx>P(x) for all x≥1 then prove that P(x)>0 for all x>1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the population of rabbits surviving at time t be governed by the differential equation dtdp(t)​=21​p(t)−200. If p(0)=100, then p(t) equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given n4<10n for a fixed positive integer n≥2, prove that (n+1)4<10n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let An​ be the area bounded by the curve y=(tanx)n and the lines x=0,y=0 and x=4π​. Prove that for n>2,An​+An−2​=n−11​ and deduce 2n+21​<An​<2n−21​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=43​−(43​)2+(43​)3+⋯+(−1)n−1(43​)n and bn​=1−an​, then find the least natural number n0​ such that bn​>an​ for all n≥n0​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that C0​−22C1​+32C2​−....+(−1)n(n+1)2Cn​=0,n>2, where Cr​=nCr​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,…,an​ are positive real numbers whose product is a fixed number c, then the minimum value of a1​+a2​+⋯+an−1​+2an​ is

If $P (1) = 0$ and $d P (x) / d x > P (x)$ for all $x \geq 1$ then prove that $P (x) > 0$ for all $x > 1$ .

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ equals:

Given $n^{4} < 1 0^{n}$ for a fixed positive integer $n \geq 2$ , prove that $(n + 1)^{4} < 1 0^{n + 1}$ .

Let $A_{n}$ be the area bounded by the curve $y = (tan x)^{n}$ and the lines $x = 0, y = 0$ and $x = \frac{π}{4}$ . Prove that for $n > 2, A_{n} + A_{n - 2} = \frac{1}{n - 1}$ and deduce $\frac{1}{2 n + 2} < A_{n} < \frac{1}{2 n - 2}$ .

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

If $P (1) = 0$ and $d P (x) / d x > P (x)$ for all $x \geq 1$ then prove that $P (x) > 0$ for all $x > 1$ .

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ equals:

Given $n^{4} < 1 0^{n}$ for a fixed positive integer $n \geq 2$ , prove that $(n + 1)^{4} < 1 0^{n + 1}$ .

Let $A_{n}$ be the area bounded by the curve $y = (tan x)^{n}$ and the lines $x = 0, y = 0$ and $x = \frac{π}{4}$ . Prove that for $n > 2, A_{n} + A_{n - 2} = \frac{1}{n - 1}$ and deduce $\frac{1}{2 n + 2} < A_{n} < \frac{1}{2 n - 2}$ .

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

Prove that $C_{0} - 2^{2} C_{1} + 3^{2} C_{2} - .... + (- 1)^{n} (n + 1)^{2} C_{n} = 0, n > 2$ , where $C_{r} =^{n} C_{r}$ .

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is