B2BSTEM
Learning Series
College Algebra
1.
Parabola Meets a Line
Consider the parabola $y = x^2 - 4$ and the line $y = x + 2$.
- Find all intersection points algebraically.
- Verify each point lies on both curves.
- Sketch both curves and label the intersection points.
x = 0.00
Hint: Set the two expressions equal: $x^2 - 4 = x + 2$. Rearrange to standard form, then factor or use the quadratic formula.
Calculus I
2.
Draining Conical Tank
Water drains from an inverted conical tank with base radius $R = 4$ m and height $H = 10$ m. The water level is dropping at $\dfrac{dh}{dt} = -0.2$ m/min when $h = 6$ m.
- Express the radius $r$ of the water surface as a function of $h$.
- Write the volume formula $V(h)$ using only $h$.
- Find $\dfrac{dV}{dt}$ when $h = 6$ m.
h = 6.0 m • r = 2.40 m
Hint: Use similar triangles to find $r = \tfrac{2}{5}h$, substitute into $V = \tfrac{1}{3}\pi r^2 h$ before differentiating — never differentiate with two variables in the formula.
Physics I
3.
Projectile from a Cliff
A ball is launched from the edge of a $40$ m cliff with initial speed $v_0 = 20.0$ m/s at angle $\theta = 30^\circ$ above horizontal. Use $g = 10$ m/s².
- Find the horizontal and vertical components of the initial velocity.
- Find the maximum height above the ground.
- Find the horizontal range when the ball hits the ground.
t = 0.00 s
Hint: Resolve $v_0$ into components. For part (c), set $y(t) = 0$ and solve the quadratic — remember the ball starts at $h = 40$ m, not $0$.
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