Math Solver

[Problem]
A right triangle has two legs both measuring 4 cm. What is the area of its circumscribed circle?

[Analysis]
This problem involves a right triangle and its circumscribed (circum) circle. For a right triangle the diameter of the circumcircle equals the length of its hypotenuse. Thus we will

• compute the hypotenuse

• find the radius as half the hypotenuse

• compute the area of the circle using $\pi r^2$

[Solution]
Step 1 Compute the hypotenuse
$h = \sqrt{4^2 + 4^2}$
$h = \sqrt{16 + 16}$
$h = \sqrt{32}$
$h = 4\sqrt{2}$

Step 2 Find the radius of the circumcircle
$r = \frac{h}{2}$
$r = \frac{4\sqrt{2}}{2}$
$r = 2\sqrt{2}$

Step 3 Compute the area of the circle
$A = \pi r^2$
$A = \pi (2\sqrt{2})^2$
$A = \pi \cdot 4 \cdot 2$
$A = 8\pi$

[Answer]
The area of the circumscribed circle is $8\pi \text{ cm}^2$.

[Problem]
On a farm, there are rabbits and turkeys totaling 41 heads and 132 legs. How many of each animal are there?

[Analysis]
This is a system of two linear equations in two variables. Let r = number of rabbits and t = number of turkeys. Use one equation for heads and one for legs, then solve by substitution (or elimination).

[Solution]
Step 1: Define variables
Let
$r$ = number of rabbits
$t$ = number of turkeys

Step 2: Write the head‐count equation
Each animal has one head, so:
$r + t = 41$

Step 3: Write the leg‐count equation
Rabbits have 4 legs and turkeys have 2 legs, so:
$4r + 2t = 132$

Step 4: Solve the first equation for one variable
From $r + t = 41$, we get
$t = 41 - r$

Step 5: Substitute into the leg equation
Replace t in $4r + 2t = 132$ with $41 - r$:
$4r + 2(41 - r) = 132$

Step 6: Expand and simplify
$4r + 82 - 2r = 132$
Combine like terms:
$2r + 82 = 132$

Step 7: Isolate r
Subtract 82 from both sides:
$2r = 50$
Divide by 2:
$r = 25$

Step 8: Find t
Use $t = 41 - r$ with $r = 25$:
$t = 41 - 25 = 16$

[Answer]
There are $25$ rabbits and $16$ turkeys.

[Problem]
See attached files

[Image]
[User uploaded an image for reference]

[Analysis]
This is a system of two linear equations in two variables. We will use the elimination method to find the values of $x$ and $y$.

[Solution]
Step 1: Add the two equations to eliminate $y$
$(2x + y) + (2x - y) = 8 + 12$
$4x = 20$

Step 2: Solve for $x$
$4x = 20 \implies x = \frac{20}{4} = 5$

Step 3: Substitute $x = 5$ into the first equation to find $y$
$2 \times 5 + y = 8$
$10 + y = 8$

Step 4: Solve for $y$
$10 + y = 8 \implies y = 8 - 10 = -2$

[Answer]
$x = 5,\; y = -2$