An object is projected upwards with a velocity on 96 feet per second from a height of 7 feet above theground is modelled by the function h(t) = -16t2 + 96t +7.A. [3 pts] When will the object reach its maximum height?B. [3 pts] Determine the maximum height that the object reached.C. [3 pts] Find the x-intercept and explain its meaning on the context of the problem.pw many seconds will the object be 90 feet above the ground? D. [3 pts] After how many seconds will the object be 90 feet above the ground?E. [2 pts] Find the y-intercept and explain its meaning on the context of the problem.ac...MacBook Air

Given: ht=-16t2+96t+7 To find: When will the object reach its maximum height? Determine the maximum height that the object reached. Find the x-intercept and explain its meaning in the context of the problem. After how many seconds will the object be 90 feet above the ground? Find the y-intercept and explain its meaning in the context of the problem. ht=-16t2+96t+7 Differentiate both sides with respect to t. h't=-162t+961+0=-32t+96 Differentiate both sides with respect to t again. h''t=-321+0=-32 To find the maximum value of ht, we need h't=0 and h''t<0. Solve h't=0 -32t+96=032t=96t=3 At t=3, h''t=-32<0 Hence, at t=3, ht has a local maximum. Answer(a): The object will reach the maximum height at t=3 seconds. To find the maximum height we will plug t=3 in ht=-16t2+96t+7. h3=-1632+963+7=151 Answer(b): The maximum height that the object reach is = 151 feet. To find the x-intercepts, we will solve ht=0 using the quadratic formula. ht=0-16t2+96t+7=0 Compare it with at2+bt+c=0. So, a=-16, b=96, c=7 Using the quadratic formula: t=-b±b2-4ac2a=-96±962-4-1672-16=-96±9664-32=−0.0720514318611 , 6.07205143186 The time can not be negative, so t=6.07205143186 seconds. Answer(c): The x-intercept is 6.07205143186 seconds. It means that at time 6.07205143186 seconds the object reaches the ground. If the object is 90 feet above the ground then ht=90 -16t2+96t+7=90-16t2+96t-83=0 Compare it with at2+bt+c=0. So, a=-16, b=96, c=-83 Using the quadratic formula: t=-b±b2-4ac2a=-96±962-4-16-832-16=-96±3904-32=1.04743758102, 4.95256241898 Hence, t=1.04743758102, 4.95256241898 seconds Answer(d): After 1.04743758102 seconds and 4.95256241898 seconds the object will be 90 feet above the ground. To find the y-intercept we will plug t=0 and we get: ht=-16t2+96t+7h0=-1602+960+7h0=7 Answer(e): The y-intercept is (0, 7). It means the initial height of the object is 7 feet.