1. A toy manufactor needs a piece of plastic in the shape of a right triangle with the longer leg 1 cm more than the shorter leg. How long should the sides of the triangle be?
a^2+b^2=c^2
x^2+(x+1)^2=(x+2)^2
x^2+x^2+2x+1= x^2+4x+4
2. Martha want to buy a rug for a room that is 11 ft wide and 18 ft. long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 144 sq. feet of carpet. What dimesons should the rug have.
(18-2x)(11-2x)=144
3.A point on the ground 60 ft from the base of the tree, the distance to the top of the tree is 4 ft more than 2 twice the height of the tree.Find the height of the tree.
h^2+60^2=(2h+4)^2
h^2+3600=(2h+4)^2
h^2+3600=(2h+4)(2h+4)
h^2+3600= 4h^2+8h+8h+16
h^2+3600=4h^2+16h+16
-h^2 -h^2
3600=3h^2+16h+16
-3600 -3600
0=3h^2+16h-3584
PLEASE HELP ME OUT IF U KNOW A STORTER WAY TO SOLVE THESE PROBLEMS
1. There are an infinte number of solutions, pick any leg, the other leg is 1cm longer. say 34cm, and 35cm.
2. correct, solve for x
3. h^2+60^2=(2h+4)^2
I would use the quadratic equation, I don't see a factor easily.
Certainly! I can help you find a shorter way to solve these problems.
1. A toy manufacturer needs a piece of plastic in the shape of a right triangle with the longer leg 1 cm more than the shorter leg. To find the lengths of the triangle's sides, let's assign a variable to represent the shorter leg. Let's say x represents the shorter leg. So the longer leg would be x + 1.
Using the Pythagorean theorem, we have:
x^2 + (x + 1)^2 = (x + 2)^2
Expanding the equation:
x^2 + (x^2 + 2x + 1) = (x^2 + 4x + 4)
Combining like terms:
2x^2 + 2x + 1 = x^2 + 4x + 4
Rearranging the equation:
x^2 - 2x - 3 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the value of x. Once we have the value of x, we can find the lengths of the triangle's sides.
2. Martha wants to buy a rug for a room that is 11 ft wide and 18 ft long. She wants to leave a uniform strip of floor around the rug. To find the dimensions of the rug, we need to determine the width of the strip.
Let's assign a variable x to represent the width of the strip. Since the rug will be smaller than the room, the dimensions of the rug will be (18 - 2x) ft by (11 - 2x) ft.
Given that the area of the rug is 144 sq. feet, we can set up an equation:
(18 - 2x)(11 - 2x) = 144
Now we can solve this quadratic equation to find the value of x. Once we have the value of x, we can substitute it back into the dimensions of the rug (18 - 2x) ft by (11 - 2x) ft to get the actual dimensions.
3. For this problem, we need to find the height of the tree. Let's assign a variable h to represent the height of the tree.
Given that a point on the ground is 60 ft from the base of the tree and the distance to the top of the tree is 4 ft more than 2 twice the height of the tree, we can set up an equation:
h^2 + 60^2 = (2h + 4)^2
Simplifying, we have:
h^2 + 3600 = 4h^2 + 16h + 16
Rearranging the equation:
3h^2 + 16h + 16 - 3600 = 0
Simplifying further:
3h^2 + 16h - 3584 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the value of h, which represents the height of the tree.
Remember, there is no one set method for solving these types of problems. Different approaches can be used, such as factoring, completing the square, or using the quadratic formula. It's important to choose a method that you find most comfortable and efficient for solving quadratic equations.