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Nutka1998 [239]
3 years ago
7

A ball is thrown vertically upward with an initial velocity of 64 feet per second. The distance s​ (in feet) of the ball from th

e ground after t seconds is s=64t-16t^2 (a) At what time t will the ball strike the​ ground? ​(b) For what time t is the ball more than 48 feet above the​ ground?
Mathematics
1 answer:
kow [346]3 years ago
6 0

Given:

The distance s​ (in feet) of the ball from the ground after t seconds is

s=64t-16t^2

To find:

(a) At what time t will the ball strike the​ ground?

(b) For what time t is the ball more than 48 feet above the​ ground?

Solution:

(a)

We have,

s=64t-16t^2

Substitute s=0, to find the time t when the ball strike the​ ground.

0=64t-16t^2

0=16t(4-t)

Using zero product property, we get

16t=0\Rightarrow t=0

4-t=0\Rightarrow t=4

Therefore, the ball strike the​ ground in initial condition (t = 0) and after 4 seconds (t = 4).

(b)

Now, s > 48, to find the time t when the ball will be more than 48 feet above the​ ground.

64t-16t^2\geq 48

0> 16t^2-64t+48

Divide both sides by 16.

0> t^2-4t+3

0> t^2-t-3t+3

0> t(t-1)-3(t-1)

0>(t-1)(t-3)

Related equation is (t-1)(t-3)=0. Zeroes are t=1,3. These two number divide the number line is three parts. (-∞,1),(1,3),(3,∞)

0>(t-1)(t-3) inequality is true for only (1,3).

It is only possible when t lies in the interval (1,3).

Therefore, the ball will be more than 48 feet above the​ ground between 1 and 3 seconds.

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According to the Vivino website, suppose the mean price for a bottle of red wine that scores 4.0 or higher on the Vivino Rating
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a) Null and alternative hypothesis

H_0: \mu=32.48\\\\H_a:\mu< 32.48

b) Test statistic t=-1.565

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NOTE: the sample size is n=65.

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d) Null and alternative hypothesis

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Test statistic t=-1.565

Critical value tc=-1.669

t>tc --> Do not reject H0

Do not reject H0. We cannot conclude that the price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.

Step-by-step explanation:

This is a hypothesis test for the population mean.

The claim is that the mean price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.

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The significance level is 0.05.

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The estimated standard error of the mean is computed using the formula:

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t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{30.15-32.48}{1.4884}=\dfrac{-2.33}{1.4884}=-1.565

The degrees of freedom for this sample size are:

df=n-1=65-1=64

This test is a left-tailed test, with 64 degrees of freedom and t=-1.565, so the P-value for this test is calculated as (using a t-table):

\text{P-value}=P(t

As the P-value (0.0612) is bigger than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

At a significance level of 0.05, there is not enough evidence to support the claim that the mean price in Providence for a bottle of red wine that scores 4.0 or higher on the Vivino Rating System is less than the population mean of $32.48.

<u>Critical value approach</u>

<u></u>

At a significance level of 0.05, for a left-tailed test, with 64 degrees of freedom, the critical value is t=-1.669.

As the test statistic is greater than the critical value, it falls in the acceptance region.

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