Write the equation of the boundary line AB of the park.
Solution and Explanation
Given:
- \[ A(0,4),\qquad B(-2,0) \]
Step 1: Find the slope.
\[ m=\frac{4-0}{0-(-2)}=\frac{4}{2}=2 \]
Step 2: Use the point-slope form.
\[ y-4=2(x-0) \]
Step 3: Simplify.
\[ \boxed{y=2x+4} \]
Answer: \(\boxed{y=2x+4}\)
Write the equation of the boundary line AC of the park.
Solution and Explanation
Given:
- \[ A(0,4),\qquad C(3,0) \]
Step 1: Find the slope.
\[ m=\frac{0-4}{3-0}=-\frac{4}{3} \]
Step 2: Use the point-slope form.
\[ y-4=-\frac{4}{3}(x-0) \]
Step 3: Simplify.
\[ \boxed{y=4-\frac{4}{3}x} \]
Answer: \(\boxed{y=4-\frac{4}{3}x}\)
Using integration, find the area of region OAC, in which
children are allowed to play cricket, football.
Solution and Explanation
Given:
- \[ A(0,4),\qquad O(0,0),\qquad C(3,0) \]
- The equation of line \(AC\) is \[ y=4-\frac{4}{3}x. \]
Using integration,
\[ \text{Area of region }OAC =\int_{0}^{3}\left(4-\frac{4}{3}x\right)dx \]
Evaluating the integral,
\[ \begin{aligned} \text{Area} & amp;=\left[4x-\frac{4}{3}\cdot\frac{x^2}{2}\right]_0^3\\ & amp;=\left[4x-\frac{2}{3}x^2\right]_0^3\\ & amp;=\left(12-6\right)-0\\ & amp;=6 \end{aligned} \]
Hence,
\[ \boxed{\text{Area of region }OAC=6\text{ square units}} \]
Using integration, find the area of region AOB.
Solution and Explanation
Given:
- \[ A(0,4),\qquad O(0,0),\qquad B(-2,0) \]
- The equation of line \(AB\) is \[ y=2x+4. \]
Using integration,
\[ \text{Area of region }AOB =\int_{-2}^{0}(2x+4)\,dx \]
Evaluating the integral,
\[ \begin{aligned} \text{Area} & amp;=\left[x^2+4x\right]_{-2}^{0}\\ & amp;=0-\left(4-8\right)\\ & amp;=4 \end{aligned} \]
Hence,
\[ \boxed{\text{Area of region }AOB=4\text{ square units}} \]
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