Can you help me set these up?

1.When 0.5cm was planed off each of the six faces of a wooden cube, its volume decreased by 169cm^3. Find its new volume.

2. The width, length, and diagonal of a rectangle are consecutive even integers. Find the integers

1. To find the new volume of the wooden cube, we can use the formula for the volume of a cube: V = s^3, where V is the volume and s is the length of one side of the cube.

Since 0.5 cm is planed off from each face, each side of the original cube is reduced by 0.5 cm.
Let's denote the original length of one side of the cube as "x". Then, the new length of one side would be (x - 0.5) cm.

According to the problem, the volume of the new cube is decreased by 169 cm^3.
So, we can set up the equation: (x - 0.5)^3 - x^3 = 169 cm^3.

Expanding the equation gives us:
(x^3 - 3x^2 * 0.5 + 3x * (0.5)^2 - (0.5)^3) - x^3 = 169 cm^3.

Simplifying the equation gives us:
-3x^2 * 0.5 + 3x * (0.5)^2 - (0.5)^3 = 169 cm^3.

Solving for x using this equation will give us the new length of one side of the cube. We can then substitute this value into the formula for the volume of a cube (V = s^3) to find the new volume.

2. Let's use algebra to solve for the consecutive even integers representing the width, length, and diagonal of the rectangle.
Let's denote the smallest even integer as "x".

Based on the problem, the even integers (width, length, diagonal) would be x, x + 2, and x + 4, respectively.

Since the diagonal of a rectangle forms a right triangle with the width and length as the legs, we can use the Pythagorean theorem to establish a relationship between the three numbers.

According to the Pythagorean theorem, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (width and length).

Thus, we can set up the equation: x^2 + (x + 2)^2 = (x + 4)^2.

Solve this quadratic equation to find the value of x. Once we have the value of x, we can find the consecutive even integers by substituting it into the expressions x, x + 2, and x + 4.

Sure! I'd be happy to help you with these questions and explain how to find the answers.

1. To find the new volume of the wooden cube after 0.5cm is planed off each face, we need to know the original volume of the cube. Let's call the original volume V.

Now, since the 0.5cm planing affects all six faces of the cube, it means that the length, width, and height of the cube will each be reduced by 0.5cm.

So, the new length, width, and height of the cube would be (L - 0.5), (W - 0.5), and (H - 0.5), where L, W, and H are the original length, width, and height of the cube, respectively.

The new volume can be calculated by multiplying these new dimensions: V_new = (L - 0.5) * (W - 0.5) * (H - 0.5).

Given that the volume decreased by 169cm^3, we can set up the equation: V - V_new = 169.

Plugging in the values, we get: V - (L - 0.5) * (W - 0.5) * (H - 0.5) = 169.

This setup will allow us to use the given information to solve for the new volume, V_new.

2. In the second question, we are given that the width, length, and diagonal of a rectangle are consecutive even integers.

Let's assume the width of the rectangle is given by W, the length by L, and the diagonal by D.

Since the width and length are consecutive even integers, we can express them as W = 2k and L = 2k + 2, where k is an integer.

The Pythagorean theorem states that the square of the diagonal of a right-angled triangle is equal to the sum of the squares of the other two sides.

So, we can write the equation as D^2 = W^2 + L^2.

Substituting the values for W and L, we get D^2 = (2k)^2 + (2k + 2)^2.

Simplifying the equation, we have D^2 = 4k^2 + 4k^2 + 8k + 4.

Combining like terms, we get D^2 = 8k^2 + 8k + 4.

Since the diagonal is also an even integer, we can express it as D = 2n, where n is an integer.

Substituting the value for D, we get (2n)^2 = 8k^2 + 8k + 4.

Simplifying further, we have 4n^2 = 8k^2 + 8k + 4.

Dividing both sides by 4, we get n^2 = 2k^2 + 2k + 1.

Now we can find values of k and n that satisfy this equation. By trying different values for k, we can find corresponding values for n.

1) original volume= s^3

new volume=s^3-169=(s-.5)^3
multipy out the cube, and then solve. The s^3 should go away.

2)in a rectangle, diagonal^2=width^2+length^2

let n be width, n+2 be length, and n+4 be diagonal

(n+4)^2=n^2+(n+2)^2
solve for n