Question

part a
a toy cannon ball is launched from a cannon on top of a platform. the equation ( h(t) = -5t^2 + 30t + 8 ) gives the height ( h ), in meters, of the ball ( t ) seconds after it is launched. what equation can be used to tell whether the ball reaches a height of 34 m?
a. ( -5t^2 + 30t + 8 = 0 )
b. ( -5t^2 + 30t + 8 = 34 )
c. ( -5t^2 + 30t + 4 + 34 = 0 )
d. ( -5t^2 + 30t + 4 = t + 34 )
part b
a toy cannon ball is launched from a cannon on top of a platform. the equation ( h(t) = -5t^2 + 30t + 8 ) gives the height ( h ), in meters, of the ball ( t ) seconds after it is launched. does the ball reach a height of 34 m?
a. yes
b. no

Explanation:

Part A

Step1: Set up the equation

We know the height function is \( h(t) = -5t^2 + 30t + 8 \). We want to find when \( h(t)=34 \), so we set up the equation \( -5t^2 + 30t + 8 = 34 \).

Step2: Rearrange to standard quadratic form

Subtract 34 from both sides to get \( -5t^2 + 30t + 8 - 34 = 0 \), which simplifies to \( -5t^2 + 30t - 26 = 0 \)? Wait, no, wait. Wait, the options: Let's check the options again. Wait, the options have \( -5t^2 + 30t + 4 + 34 = 0 \)? No, wait, the original function is \( h(t)=-5t^2 + 30t + 8 \). So setting \( h(t)=34 \) gives \( -5t^2 + 30t + 8 = 34 \), then subtract 34: \( -5t^2 + 30t + 8 - 34 = 0 \) → \( -5t^2 + 30t - 26 = 0 \). But the options: Wait, maybe a typo? Wait, option C is \( -5t^2 + 30t + 4 + 34 = 0 \)? No, wait, maybe the original function was \( -5t^2 + 30t + 4 \)? Wait, the image shows the function as \( h(t)=-5t^2 + 30t + 8 \)? Wait, no, looking at Part B: "The equation \( h(t) = -5t^2 + 30t + 8 \) gives the height...". So Part A: set \( h(t)=34 \), so \( -5t^2 + 30t + 8 = 34 \), then subtract 34: \( -5t^2 + 30t + 8 - 34 = 0 \) → \( -5t^2 + 30t - 26 = 0 \). But the options: Let's check the options again. Option C: \( -5t^2 + 30t + 4 + 34 = 0 \)? No, maybe a mistake in the options, but wait, maybe the function in Part A is \( -5t^2 + 30t + 4 \)? Wait, no, Part B says \( h(t)=-5t^2 + 30t + 8 \). Wait, maybe the options have a typo, but among the options, option C is \( -5t^2 + 30t + 4 + 34 = 0 \)? No, wait, option C is \( -5t^2 + 30t + 4 + 34 = 0 \)? No, the options are:

A. \( -5t^2 + 30t + 8 = 0 \)

B. \( -5t^2 + 30t + 8 = 34 \)

C. \( -5t^2 + 30t + 4 + 34 = 0 \) → no, wait, maybe the function is \( -5t^2 + 30t + 4 \), then setting to 34: \( -5t^2 + 30t + 4 = 34 \), then \( -5t^2 + 30t + 4 - 34 = 0 \) → \( -5t^2 + 30t - 30 = 0 \)? No. Wait, maybe the correct equation is to set \( h(t)=34 \), so \( -5t^2 + 30t + 8 = 34 \), which is option B: \( -5t^2 + 30t + 8 = 34 \). Wait, option B is \( -5t^2 + 30t + 8 = 34 \), which is the equation we set up. So Part A answer is B? Wait, no, the question is "what equation can be used to tell whether the ball reaches a height of 34 m". So we set \( h(t)=34 \), so the equation is \( -5t^2 + 30t + 8 = 34 \), which is option B.

Step3: Confirm

So the equation to solve is \( -5t^2 + 30t + 8 = 34 \), which is option B.

Step1: Set up the equation

We have \( h(t) = -5t^2 + 30t + 8 \), and we want to check if \( h(t)=34 \) has a solution. So set \( -5t^2 + 30t + 8 = 34 \), then rearrange to \( -5t^2 + 30t - 26 = 0 \).

Step2: Calculate the discriminant

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is \( D = b^2 - 4ac \). Here, \( a = -5 \), \( b = 30 \), \( c = -26 \). So \( D = 30^2 - 4(-5)(-26) = 900 - 4(130) = 900 - 520 = 380 \).

Step3: Analyze the discriminant

Since \( D = 380 > 0 \), the quadratic equation has two real solutions, which means the ball does reach a height of 34 m at some time \( t \).

Answer:

(Part A): B. \( -5t^2 + 30t + 8 = 34 \)

Part B