1) A box with a square base has a total surface area of 12m^2. Express the volume of the box as a function of the base edge.
2) A 10 foot tall lamppost causes a child 4 feet tall to cast a shadow of x feet on the ground. If the distance along the ground from the child to the lamppost is d meters, express x as a function.
let the side of the square base be x m
let the height be y m
I will assume the box has a lid so the surface area is made up of 2x^ + 4xy
then 2x^2 + 4xy = 12
4xy = 12-2x^2
y = (6-x^2)/(2x)
volume = x^2y
= x^2(6-x^2)/(2x)
= x(6-x^2)/2
for the second, you have similar triangles, so set up a ratio
4/x = 10/(x+d)
10x = 4x + 4d
take it from there
1) To find the volume of the box as a function of the base edge, we need to first find the height of the box.
The total surface area of the box is given as 12m^2. This consists of the area of the square base plus the area of the four sides.
The area of the square base is equal to the base edge squared, so it's x^2 square meters.
The area of each side of the box is equal to the base edge multiplied by the height, so each side has an area of 2xh square meters (since there are four sides).
The total surface area equation can now be written as: x^2 + 4(2xh) = 12.
Simplifying the equation gives: x^2 + 8xh = 12.
To express the volume of the box as a function of the base edge, we need to solve for the height h. Rearranging the equation gives: h = (12 - x^2) / 8x.
The volume V of the box is given by the formula V = x^2h. Substituting the value of h above, we get: V = x^2((12 - x^2) / 8x).
Simplifying the expression gives the volume as a function of the base edge x: V(x) = (12x - x^3) / 8.
2) To express x (the length of the shadow) as a function, we need to relate it to the height of the lamppost and the distance from the child to the lamppost.
From the given information, we know that the height of the lamppost is 10 feet and the height of the child is 4 feet.
Let's consider similar triangles. The height of the lamppost and the child's height create similar triangles with the shadow on the ground. The ratio of the child's height to the shadow length is the same as the ratio of the lamppost's height to the distance from the child to the lamppost.
So, we have the following proportion: (height of child) / x = (height of lamppost) / d.
Substituting the given values, we get: 4 / x = 10 / d.
To express x as a function, we isolate it by cross-multiplying and solving for x: 4d = 10x.
Dividing both sides of the equation by 10, we get: x = (4d) / 10.
Simplifying the expression further gives x as a function of the distance d: x(d) = (2d) / 5.