Exponential growth & decay
A town has a population of 10,000 people and is growing at an annual rate of 5%. What will the population be in 5 years?
The population will be approximately 12,765.
Write the equation of a quadratic in standard form if the vertex is at (2, 3) and it opens upwards with a width of 1.
The equation in standard form is \( y = (x - 2)^2 + 3 \) or any equivalent quadratic function that meets the vertex and opening requirements.
What is the square root of 64?
8
A rectangle has a length of 10 cm and a width of 5 cm. What is the area and perimeter of the rectangle
Area = Length × Width = 10 cm × 5 cm = 50 cm²
Perimeter = 2(Length + Width) = 2(10 cm + 5 cm) = 30 cm
Find the slope of the line that passes through the points (2, 3) and (4, 7).
4,6 and 8,14
A certain radioactive material has a half-life of 10 years. If you start with 80 grams of this material, how much will remain after 30 years?
There will be 10 grams remaining.
Convert the following standard form quadratic equation to vertex form: ( y = 2x^2 - 8x + 5 )
( y = 2(x^2 - 4x) + 5 \) → \( y = 2((x - 2)^2 - 4) + 5 \) → \( y = 2(x - 2)^2 - 8 + 5 \) → \( y = 2(x - 2)^2 - 3 \)
Simplify the square root of 49.
7
A triangle has a base of 8 m and a height of 5 m. What is the area of the triangle? (Assume the triangle is a right triangle)
- Area = (Base × Height) / 2 = (8 m × 5 m) / 2 = 20 m²
Write the equation of the line in slope-intercept form that has a slope of 3 and passes through the point (1, 2).
2 = 3(1) + b \implies b = 2 - 3 = -1 \] The equation is \( y = 3x - 1 \).
You invest $1,000 in a savings account that offers a 3% annual compound interest rate. How much money will you have in the account after 4 years?
You will have approximately $1,125.51 in the account.
Determine the x-intercepts of the quadratic equation in standard form
( y = 0 \): \( 0 = x^2 - 4x - 5 \). ( (x - 5)(x + 1) = 0 \) → x-intercepts are \( x = 5 \) and \( x = -1 \).
Calculate the square root of 144.
12
A circle has a radius of 7 inches. What is the area and circumference of the circle? (Use π = 3.14).
Area = π × Radius² = 3.14 × (7 in)² = 3.14 × 49 in² = 153.86 in²
Circumference = 2 × π × Radius = 2 × 3.14 × 7 in = 43.96 in
Determine whether the following two lines are parallel, perpendicular, or neither:
Line 1: \( 3y = -2x + 6 \) → \( y = -\frac{2}{3}x + 2 \) (slope = -2/3)
Line 2: \( -2y = -3x + 4 \) → \( y = \frac{3}{2}x - 2 \) (slope = 3/2)
A culture of bacteria doubles in size every 3 hours. If you start with 200 bacteria, how many will there be after 12 hours?
There will be 3,200 bacteria after 12 hours.
If \( f(x) = 3x^2 + 12x + 9 \), find the vertex of the parabola represented by the quadratic function.
(-2, -3).
Find the value of the square root of 225.
15
A square has a side length of 4 ft. What is the area and perimeter of the square?
Area = Side × Side = 4 ft × 4 ft = 16 ft²
Perimeter = 4 × Side = 4 × 4 ft = 16 ft
Find the x-intercept of the linear equation \( 4x - 5y = 20 \).
To find the x-intercept, set \( y = 0 \): \[ 4x - 5(0) = 20 \implies 4x = 20 \implies x = 5 \] The x-intercept is (5, 0).
A car is valued at $20,000 and depreciates at a rate of 15% per year. What will be the car’s value after 4 years?
The car’s value will be approximately $10,440 after 4 years.
Identify the y-intercept of the quadratic function given by the equation: \( y = -2x^2 + 4x + 1 \).
The y-intercept is found by evaluating when \( x = 0 \): \( y = -2(0)^2 + 4(0) + 1 = 1 \).
Thus, the y-intercept is at (0, 1)
What is the square root of 36?
6
A trapezoid has bases of 6 m and 10 m, and a height of 4 m. What is the area of the trapezoid?
Area = (Base1 + Base2) / 2 × Height = (6 m + 10 m) / 2 × 4 m = 16 m²
If the function \( f(x) = 5x - 4 \), what is the value of \( f(2) \)?
Substituting \( x = 2 \) into the function: \[ f(2) = 5(2) - 4 = 10 - 4 = 6 \] \( f(2) = 6 \).