1. Express the area of a square as a function of its diagonal.
2. The height of a cylinder is half the diameter. Express the total surface area as a function of the height. SA = 2πh + 2πr^2
1. To express the area of a square as a function of its diagonal, we need to find a relationship between the diagonal and the side length of the square.
Let's assume that the side length of the square is represented by the variable "s", and the diagonal is represented by the variable "d".
In a square, the diagonal divides the square into two congruent right-angled triangles. By applying the Pythagorean theorem, we can find the relationship between the side length and the diagonal.
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (two sides of the square).
So, we have:
d^2 = s^2 + s^2
d^2 = 2s^2
To find the area of the square, we can use the formula A = s^2.
Now, we need to eliminate the "s" variable from the equation using the relationship we found earlier.
From the previous equation, we have:
d^2 = 2s^2
Dividing both sides of the equation by 2, we get:
s^2 = (1/2)d^2
Taking the square root of both sides, we can express the side length (s) as:
s = √((1/2)d^2)
Finally, we can substitute this value for "s" into the formula for the area of a square:
A = s^2 = (√((1/2)d^2))^2
A = (1/2)d^2
Therefore, the area of a square can be expressed as a function of its diagonal (d) using the equation:
A = (1/2)d^2
2. To express the total surface area (SA) of a cylinder as a function of its height (h), we need to consider the relationship between the height and the radius (r) of the cylinder.
Given that the height is half the diameter, we can express the height as:
h = r/2
The total surface area of a cylinder is given by the formula:
SA = 2πr^2 + 2πrh
Substituting the value of h that we found earlier, we have:
SA = 2πr^2 + 2πr(r/2)
Simplifying the equation, we get:
SA = 2πr^2 + πr^2
SA = 3πr^2
Therefore, the total surface area of a cylinder can be expressed as a function of its height (h) using the equation:
SA = 3πr^2, where h = r/2
1. To express the area of a square as a function of its diagonal, we need to establish a relationship between the diagonal and the side length of the square.
Let's assume that the diagonal of the square is denoted by "d" and the side length by "s". In a square, the diagonal divides the square into two right-angled triangles, where the hypotenuse is "d" and the two sides are "s".
Using the Pythagorean theorem, we can relate the diagonal and side length as follows:
d^2 = s^2 + s^2
d^2 = 2s^2
Now, solving for the side length:
s^2 = d^2 / 2
s = √(d^2 / 2)
s = d / √2
Finally, we can express the area of the square in terms of the diagonal:
Area = s^2
Area = (d / √2)^2
Area = d^2 / 2
Therefore, the area of a square can be expressed as a function of its diagonal using the formula: Area = d^2 / 2.
2. To express the total surface area of a cylinder as a function of its height, we need to understand the components that contribute to the surface area.
A cylinder consists of two circular bases and a curved surface. The height (h) of the cylinder is half the diameter (d), which means the radius (r) is equal to d/2.
The formula for the total surface area (SA) of a cylinder is:
SA = 2πrh + 2πr^2
Substituting the value of the radius (r) from the given height:
r = d/2
SA = 2π(d/2)h + 2π(d/2)^2
SA = πdh + π(d^2/4)
SA = π(dh + d^2/4)
Therefore, we can express the total surface area of a cylinder as a function of the height using the formula: SA = π(dh + d^2/4).