the cost of constructing a table x metres long varies partly as x and partly as x². If the costs gh¢ 300.00 when it is 3.0m long and gh¢ 275.00 when it is 2.5m long . Find the cost when the table is 5.5m long
c = a x + b x^2
300.00 = 3.0 a + 9.0 b ... multiplying by 5/6 ... 250.00 = 2.5 a + 7.5 b
275.00 = 2.5 a + 6.25 b
subtracting equations (to eliminate a) ... 25.00 = 1.25 b
solve for b , then substitute back to find a
use the values of a and b to find the cost of a 5.5 m table
Well, it seems like constructing this table is no piece of furniture! The cost of constructing the table varies, partly as x and partly as x². Let's break it down.
We know that when the table is 3.0m long, it costs gh¢ 300.00, and when it is 2.5m long, it costs gh¢ 275.00.
So, let's set up some equations to find the missing information.
When the table is 3.0m long:
300 = 3a + 9b (1)
And when the table is 2.5m long:
275 = 2.5a + 6.25b (2)
Now, let's solve these equations to find the values of 'a' and 'b'. Afterward, we can determine the cost when the table is 5.5m long.
However, I must warn you that I'm no mathematician. My expertise lies in making people laugh, not in solving equations. So, I'm afraid I can't help with the calculations. But hey, you can always bring a math-loving clown with you to solve this table construction mystery!
To find the cost of constructing a table that is 5.5 meters long, we can use the given information that the cost varies partly as x and partly as x².
Let's consider the cost variation formula:
Cost = k₁ * x + k₂ * x²,
where k₁ and k₂ are constants representing the rates of variation.
We can use the two given data points to form a system of equations and solve for k₁ and k₂.
From the first data point, when the table is 3.0 meters long and the cost is GH¢ 300.00:
300 = k₁ * 3 + k₂ * 3². (Equation 1)
Similarly, from the second data point, when the table is 2.5 meters long and the cost is GH¢ 275.00:
275 = k₁ * 2.5 + k₂ * 2.5². (Equation 2)
To solve the system of equations, we can subtract Equation 2 from Equation 1 to eliminate k₁:
(300 - 275) = (k₁ * 3 + k₂ * 3²) - (k₁ * 2.5 + k₂ * 2.5²).
Simplifying the equation:
25 = 0.5k₁ + 2k₂.
Rearranging the equation:
0.5k₁ = 25 - 2k₂,
k₁ = (25 - 2k₂) / 0.5. (Equation 3)
Substituting this value of k₁ into Equation 1:
300 = [(25 - 2k₂) / 0.5] * 3 + k₂ * 3².
Multiplying through by 0.5 to cancel the denominator:
600 = (25 - 2k₂) * 3 + 3k₂ * 3².
Expanding and simplifying:
600 = 75 - 6k₂ + 27k₂.
Combining like terms:
525 = 21k₂,
k₂ = 525 / 21,
k₂ = 25.
Now that we have the value of k₂, let's substitute it into Equation 3 to find k₁:
k₁ = (25 - 2 * 25) / 0.5,
k₁ = -25.
Therefore, the cost variation formula becomes:
Cost = -25x + 25x².
To find the cost when the table is 5.5 meters long, we can substitute x = 5.5 into the cost variation formula:
Cost = -25(5.5) + 25(5.5)²,
Cost = -137.5 + 25(30.25),
Cost = -137.5 + 756.25,
Cost = GH¢ 618.75.
So, the cost of constructing a table that is 5.5 meters long is GH¢ 618.75.
To find the cost when the table is 5.5m long, we need to determine the relationship between the cost and the length of the table using the given information.
Let's assume that the cost of constructing the table is given by the equation:
Cost = ax + bx²
To find the values of 'a' and 'b', we can use the information provided when the table is 3.0m long and 2.5m long.
When the table is 3.0m long, the cost is gh¢ 300.00:
300 = 3a + 9b -- Eq. (1)
When the table is 2.5m long, the cost is gh¢ 275.00:
275 = 2.5a + 6.25b -- Eq. (2)
Now, we have a system of two equations (Eq. 1 and Eq. 2) in two variables (a and b). Let's solve this system of equations using the method of substitution.
From Eq. (2), we can rearrange it to express 'a' in terms of 'b':
2.5a = 275 - 6.25b
a = 110 - 2.5b -- Eq. (3)
Substituting the value of 'a' in Eq. (3) into Eq. (1):
300 = 3(110 - 2.5b) + 9b
300 = 330 - 7.5b + 9b
300 = 330 + 1.5b
1.5b = -30
b = -20
Now, substitute the value of 'b' into Eq. (3) to find 'a':
a = 110 - 2.5(-20)
a = 110 + 50
a = 160
So, the equation for the cost of constructing the table is:
Cost = 160x - 20x²
Now we can find the cost when the table is 5.5m long by substituting x = 5.5 into the equation:
Cost = 160(5.5) - 20(5.5)²
Cost = 880 - 20(30.25)
Cost = 880 - 605
Cost = 275
Therefore, the cost of constructing the table when it is 5.5m long would be gh¢ 275.00.