Question

there is a positive and negative x - intercept for this function. what is the positive x - intercept? what does this x - intercept represent in this situation? the positive x - intercept is

. this means

diver jumps diving board
a diver jumps from a diving board into a pool at his swim club. his height h(t) in meters t seconds after the dive can be estimated using the equation h(t)= - 4.9t² + 4.3t + 10.

Explanation:

Step1: Recall x - intercept definition

Set \(h(t)=0\), so \( - 4.9t^{2}+4.3t + 10=0\).

Step2: Use quadratic formula

The quadratic formula for \(ax^{2}+bx + c = 0\) is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here \(a=-4.9\), \(b = 4.3\), \(c = 10\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(4.3)^{2}-4\times(-4.9)\times10=18.49 + 196=214.49\). Then \(t=\frac{-4.3\pm\sqrt{214.49}}{2\times(-4.9)}=\frac{-4.3\pm14.645}{-9.8}\).

Step3: Find the two solutions

\(t_1=\frac{-4.3 + 14.645}{-9.8}=\frac{10.345}{-9.8}\approx - 1.056\) and \(t_2=\frac{-4.3-14.645}{-9.8}=\frac{-18.945}{-9.8}\approx1.933\).

Step4: Interpret the results

The positive \(x\) - intercept is \(t\approx1.933\). The negative \(x\) - intercept has no physical meaning in the context of time (since time \(t\) cannot be negative in the situation of a diver diving from a board). The positive \(x\) - intercept represents the time when the diver hits the water surface.

Answer:

The positive \(x\) - intercept is approximately \(1.933\). This means the time when the diver hits the water.