Question

the students also built the discovery, a third underwater drone. its path can be modeled using this function, where x is the horizontal distance, in miles, from the launch site and d(x) is the depth below the surface of the water, in meters. d(x) = 0.4x² - 20x + 70 rewrite the equation in vertex form. then use the vertex form to determine which two statements are true. the drone reaches its minimum when its 50 miles from the launch site. the drones minimum is a depth of -70 meters. the drone reaches its minimum when its 25 miles from the launch site. the drones minimum is a depth of -430 meters. the drone reaches its minimum when its 46 miles from the launch site. the drones minimum is a depth of -180 meters.

Explanation:

Step1: Factor out the coefficient of \(x^2\)

Given \(d(x) = 0.4x^2 - 20x + 70\), factor out \(0.4\) from the first two terms:
\(d(x) = 0.4(x^2 - 50x) + 70\)

Step2: Complete the square inside the parentheses

For the quadratic \(x^2 - 50x\), the formula to complete the square is \(x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2\). Here, \(b = -50\), so \(\frac{b}{2} = -25\), and \((\frac{b}{2})^2 = 625\).
Add and subtract \(625\) inside the parentheses (multiplying by \(0.4\) to maintain equality):
\(d(x) = 0.4[(x^2 - 50x + 625) - 625] + 70\)

Step3: Simplify to vertex form

Simplify the expression:
\(d(x) = 0.4(x - 25)^2 - 0.4(625) + 70\)
Calculate \(0.4(625) = 250\), so:
\(d(x) = 0.4(x - 25)^2 - 250 + 70\)
\(d(x) = 0.4(x - 25)^2 - 180\)

Step4: Analyze the vertex form

The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. For \(d(x) = 0.4(x - 25)^2 - 180\), \(a = 0.4 > 0\) (so the parabola opens upward, and the vertex is a minimum).

• The \(x\)-coordinate of the vertex (horizontal distance) is \(h = 25\) miles.
• The \(y\)-coordinate of the vertex (minimum depth) is \(k = -180\) meters.

Answer:

The two true statements are:

• The drone reaches its minimum when it’s 25 miles from the launch site.
• The drone’s minimum is a depth of -180 meters.