Set up an algebraic equation and then solve the following problems.
An integer is 3 less than 5 times another. If the product of the two integers is 36, then find the integers.
The width of a rectangle is 5 units less than the length. If the area is 150 square units, then find the dimensions of the rectangle.
The length of a rectangle is 4 inches more than its width. The area of the rectangle is equal to 5 inches more than 2 times the perimeter. Find the length and width of the rectangle.
The height of a projectile launched upward at a speed of 32 feet/second from a height of 128 feet is given by the function h(t)=-16t^2+32t+128. How long will it take the projectile to hit the ground?
The height of an object dropped from the top of a 144-foot building is given by h(t)=-16t^2+144. How long will it take the object to hit the ground?
Here are the solutions to the problems, along with the algebraic equations used:
1. Integer Problem:
-
Equations:
- Let x be the first integer.
- The second integer is 5x - 3.
- Their product: x(5x - 3) = 36
-
Solution:
- 5x^2 - 3x - 36 = 0
- Factoring: (5x + 9)(x - 4) = 0
- x = 4 or x = -9/5
- The integers are 4 and 17 (discarding the negative value).
2. Rectangle Dimensions:
-
Equations:
- Let l be the length.
- The width is l - 5.
- Area: l(l - 5) = 150
-
Solution:
- l^2 - 5l - 150 = 0
- Factoring: (l - 15)(l + 10) = 0
- l = 15 or l = -10 (discarding the negative value).
- The dimensions are 15 units by 10 units.
3. Rectangle with Area and Perimeter:
-
Equations:
- Let w be the width.
- The length is w + 4.
- Area: w(w + 4)
- Perimeter: 2(2w + 4)
- Given: w(w + 4) = 2(2w + 4) + 5
-
Solution:
- w^2 + 4w = 4w + 13
- w^2 = 13
- w = √13 (approximately 3.61 inches)
- The length is 7.61 inches.
4. Projectile Height:
-
Equation:
- h(t) = -16t^2 + 32t + 128
- The projectile hits the ground when h(t) = 0.
-
Solution:
- -16t^2 + 32t + 128 = 0
- Divide by -4: 4t^2 - 8t - 32 = 0
- Factoring: (2t - 8)(2t + 4) = 0
- t = 4 or t = -2 (discarding the negative value).
- It takes 4 seconds to hit the ground.
5. Dropped Object Height:
-
Equation:
- h(t) = -16t^2 + 144
- The object hits the ground when h(t) = 0.
-
Solution:
- -16t^2 + 144 = 0
- Divide by -16: t^2 - 9 = 0
- Factoring: (t - 3)(t + 3) = 0
- t = 3 or t = -3 (discarding the negative value).
- It takes 3 seconds to hit the ground.