A quantity p varies partly t and partly as the square of t, when t=20, p=45 and when t=24, p=60. (i) Express p in terms of t.(ii) Find p when t=32

The problem states that p varies partly as t and partly as the square of t. This means we can express p as:

p = kt + ct²

where k and c are constants that we need to find.

Given that when t=20, p=45 and when t=24, p=60, we can form two equations:

45 = 20k + 400c ...(1) 60 = 24k + 576c ...(2)

We can solve these two equations simultaneously to find the values of k and c.

Subtract equation (1) from equation (2):

15 = 4k + 176c

Divide through by 4:

3.75 = k + 44c ...(3)

Substitute equation (3) into equation (1):

45 = 20(3.75 - 44c

The problem states that p varies partly as t and partly as the square of t. This means we can express p as:

p = kt + ct²

where k and c are constants that we need to find.

Given that when t=20, p=45 and when t=24, p=60, we can form two equations:

45 = 20k + 400c ...(1) 60 = 24k + 576c ...(2)

We can solve these two equations simultaneously to find the values of k and c.

Subtract equation (1) from equation (2):

15 = 4k + 176c

Divide through by 4:

3.75 = k + 44c ...(3)

Substitute equation (3) into equation (1):

45 = 20(3.75 - 44c) + 400c 45 = 75 - 880c + 400c 45 = 75 - 480c 480c = 30 c = 30/480 = 0.0625

Substitute c = 0.0625 into equation (3):

3.75 = k + 44(0.0625) 3.75 = k + 2.75 k = 3.75 - 2.75 = 1

So, the equation of p in terms of t is:

p = t + 0.0625t²

(ii) To find p when t=32, substitute t=32 into the equation:

p = 32 + 0.0625(32)² p = 32 + 0.0625(1024) p = 32 + 64 p = 96

The problem states that p varies partly as t and partly as the square of t. This means that we can express p as the sum of two terms, one that varies directly as t and one that varies as the square of t. We can write this relationship as:

p = kt + lt^2

where k and l are constants that we need to determine.

Given that when t=20, p=45 and when t=24, p=60, we can set up the following system of equations to solve for k and l:

45 = 20k + 400l 60 = 24k + 576l

We can solve this system of equations by substitution or elimination to find the values of k and l.

Subtract the first equation from the second to get:

15 = 4k + 176l

Divide through by 4 to get:

3.75 = k + 44l

Substitute this into the first equation to get:

45 = 20(3.75 - 44l) + 400l 45 = 75 - 880l + 400l 45 = 75 - 480l -30 = -480l l = 0.0625

Substitute l = 0.0625 into

The problem states that p varies partly as t and partly as the square of t. This means that we can express p as the sum of two terms, one that varies directly with t and one that varies with the square of t. We can write this relationship as:

p = kt + lt^2

where k and l are constants that we need to find.

Given that when t=20, p=45 and when t=24, p=60, we can set up the following system of equations to solve for k and l:

45 = 20k + 400l 60 = 24k + 576l

We can solve this system of equations by substitution or elimination to find the values of k and l.

Subtract the first equation from the second to get:

15 = 4k + 176l

Divide through by 4 to get:

3.75 = k + 44l

Substitute this into the first equation to get:

45 = 20(3.75 - 44l) + 400l 45 = 75 - 880l + 400l 45 = 75 - 480l

Solving for l gives:

l = (75 - 45) / 480 = 0.0625

Substitute l = 0.0625 into the equation 3.75 = k + 44l to get:

k = 3.75 - 44*0.0625 = 0.75

So, the equation that expresses p in terms of t is:

p = 0.75t + 0.0625t^2

To

The problem states that p varies partly as t and partly as the square of t. This means that we can express p as the sum of two terms, one that varies directly as t and one that varies as the square of t. We can write this relationship as:

p = kt + lt^2

where k and l are constants that we need to find.

We are given two pairs of values for p and t:

When t = 20, p = 45 When t = 24, p = 60

We can substitute these values into our equation to get two equations in k and l:

45 = 20k + 400l

The problem states that p varies partly as t and partly as the square of t. This means that we can express p as the sum of two terms, one that varies directly as t and one that varies as the square of t. We can write this relationship as:

p = kt + lt^2

where k and l are constants that we need to find.

We are given two pairs of values for p and t:

When t = 20, p = 45 When t = 24, p = 60

We can substitute these values into our equation to get two equations in two unknowns:

45 = 20k + 400l 60 = 24k + 576l

We can solve this system of equations to find the values of k and l.

Subtract the first equation from the second to get:

15 = 4k + 176l

Solve this equation for k:

k = (15 - 176l) / 4

Substitute this value for k into the first equation:

45 = 20((15 - 176l) / 4) + 400l 45 = 75 - 880l + 400l 45 = 75 - 480l

Solve this equation for l:

l = (75 - 45) / 480 l = 0.0625

Substitute l = 0.0625 into the equation for k:

k = (15 - 176(0.0625)) / 4 k = 1.25

So, the equation that expresses p in terms of t is:

p = 1.25t + 0.0625t^2

To find p when t = 32, substitute t = 32 into this equation:

p = 1.25(32) + 0.0625(32)^2 p = 40 + 64 p = 104