Solve the math problems provided in the image.

Steps to Solve

7. Ratio of Circumferences to Radii

The circumference of a circle is given by $C = 2\pi r$, where $r$ is the radius. If we have two circles with circumferences $C_1$ and $C_2$ and radii $r_1$ and $r_2$ respectively, then $C_1 = 2\pi r_1$ and $C_2 = 2\pi r_2$. The ratio of their circumferences is given as 5:3, so we have $\frac{C_1}{C_2} = \frac{5}{3}$. $$ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} $$ Since $\frac{C_1}{C_2} = \frac{5}{3}$, then $\frac{r_1}{r_2} = \frac{5}{3}$. Therefore, the ratio of their radii is 5:3.

8. Finding the value of x in a triangle's perimeter

The perimeter of a triangle is the sum of the lengths of its three sides. We are given that the perimeter of $\triangle ABC$ is 50 cm, and the lengths of its sides are AB = 15 cm, BC = 10 cm, and AC = (x + 4) cm. Therefore, we can write the equation: $AB + BC + AC = 50$ $15 + 10 + (x + 4) = 50$ $25 + x + 4 = 50$ $29 + x = 50$ $x = 50 - 29$ $x = 21$

9. Finding the length of a rectangular park

The perimeter of a rectangle is given by $P = 2(l + b)$, where $l$ is the length and $b$ is the breadth. The breadth of the rectangular park is given as 48 m. The length of the wire required for fencing it is equal to the perimeter of the park, which is 3 km 72 m. First, convert the length of the wire to meters: 3 km = 3000 m, so 3 km 72 m = 3000 m + 72 m = 3072 m. Now we can set up the equation: $2(l + 48) = 3072$ $l + 48 = \frac{3072}{2}$ $l + 48 = 1536$ $l = 1536 - 48$ $l = 1488$ m

10. Finding the breadth of a rectangle given the perimeter of a square and the ratio of the rectangle's sides

The perimeter of the square is 120 m. Let the side of the square be $s$. Then $4s = 120$, so $s = \frac{120}{4} = 30$ m. The perimeter of the rectangle is half the perimeter of the square, so the perimeter of the rectangle is $\frac{120}{2} = 60$ m. The length and breadth of the rectangle are in the ratio 3:2. Let the length be $3y$ and the breadth be $2y$. The perimeter of the rectangle is $2(3y + 2y) = 60$, so $2(5y) = 60$, which means $10y = 60$. Thus, $y = \frac{60}{10} = 6$. The breadth of the rectangle is $2y = 2(6) = 12$ m.

11. Finding the difference between the radii of two concentric circles Let the radius of the outer circle be $R$ and the radius of the inner circle be $r$. The circumference of the outer circle is $132$ cm, so $2\pi R = 132$. The circumference of the inner circle is $88$ cm, so $2\pi r = 88$. We want to find $R - r$. $2\pi R - 2\pi r = 132 - 88$ $2\pi (R - r) = 44$ $R - r = \frac{44}{2\pi} = \frac{22}{\pi}$. Using $\pi = \frac{22}{7}$, we have $R - r = \frac{22}{\frac{22}{7}} = 22 \cdot \frac{7}{22} = 7$ cm.

12. Finding the radius of a circle formed by reshaping a rectangle

The length of the rectangle is 8.9 cm, and the breadth is 54 mm. First, convert the length to mm: $8.9 \text{ cm} = 8.9 \times 10 \text{ mm} = 89 \text{ mm}$. The perimeter of the rectangle is $2(l + b) = 2(89 + 54) = 2(143) = 286 \text{ mm}$. When the wire is reshaped into a circle, the circumference of the circle is equal to the perimeter of the rectangle. So, $2\pi r = 286$, where $r$ is the radius of the circle. $r = \frac{286}{2\pi} = \frac{143}{\pi}$. Using $\pi = \frac{22}{7}$, $r = \frac{143}{\frac{22}{7}} = \frac{143 \times 7}{22} = \frac{13 \times 7}{2} = \frac{91}{2} = 45.5 \text{ mm}$.

13. Finding the diameter of a circular ring formed by rebending a square

The side of the square is 11 cm. The perimeter of the square is $4 \times 11 = 44$ cm. When the wire is rebent to form a circular ring, the circumference of the circle equals the perimeter of the square. Thus, $2\pi r = 44$, where $r$ is the radius of the circle. The diameter of the ring is $2r$, so we want to find $2r$. Since $2\pi r = 44$, $2r = \frac{44}{\pi}$. Using $\pi = \frac{22}{7}$, we have $2r = \frac{44}{\frac{22}{7}} = \frac{44 \times 7}{22} = 2 \times 7 = 14$ cm.

14. Finding the radius of a circle formed by rebending a rectangle

The length of the rectangle is 40 cm and the breadth is 26 cm. The perimeter of the rectangle is $2(40 + 26) = 2(66) = 132$ cm. When the wire is rebent into a circle, the circumference of the circle is equal to the perimeter of the rectangle. So, $2\pi r = 132$, where $r$ is the radius of the circle. $r = \frac{132}{2\pi} = \frac{66}{\pi}$. Using $\pi = \frac{22}{7}$, $r = \frac{66}{\frac{22}{7}} = \frac{66 \times 7}{22} = 3 \times 7 = 21$ cm.