Given the curve y = x2 - 2x + 6, find the coordinate of the point on the curve at which the gradient is 4…

Question

Given the curve y = x2 - 2x + 6, find the coordinate of the point on the curve at which the gradient is 4. Also, Mary has some money in two denominations only. Fifty shilling notes and twenty shilling

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Here are the solutions to the problems:

Given the curve $y = x^2 - 2x + 6$ , find the coordinate of the point on the curve at which the gradient is 4.

Find the derivative of the curve to get the gradient function.
The gradient, $\frac{dy}{dx}$ , is found by differentiating the equation $y = x^2 - 2x + 6$ with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(x^2 - 2x + 6)$ $\frac{dy}{dx} = 2x - 2$
Set the gradient equal to 4 and solve for $x$ .
$2x - 2 = 4$ $2x = 4 + 2$ $2x = 6$ $x = \frac{6}{2}$ $x = 3$
Substitute the value of $x$ back into the original equation to find the corresponding $y$ -coordinate.
$y = (3)^2 - 2(3) + 6$ $y = 9 - 6 + 6$ $y = 9$
The coordinate of the point is $\boxed{(3, 9)}$ .

13. Mary has some money in two denominations only. Fifty shilling notes and twenty shilling coins. She has three times as many fifty shilling notes as twenty shilling coins. If altogether she has KSh 3400. Find the number of fifty shilling notes and twenty shilling coins.
Define variables.
Let $N_{50}$ be the number of fifty shilling notes.
Let $N_{20}$ be the number of twenty shilling coins.
Formulate equations based on the given information.
From "She has three times as many fifty shilling notes as twenty shilling coins":
$N_{50} = 3N_{20} \quad \text{(Equation 1)}$
From "If altogether she has KSh 3400":
$50N_{50} + 20N_{20} = 3400 \quad \text{(Equation 2)}$
Substitute Equation 1 into Equation 2.
Substitute $N_{50} = 3N_{20}$ into $50N_{50} + 20N_{20} = 3400$ :
$50(3N_{20}) + 20N_{20} = 3400$ $150N_{20} + 20N_{20} = 3400$ $170N_{20} = 3400$
Solve for $N_{20}$ .
$N_{20} = \frac{3400}{170}$ $N_{20} = 20$
So, Mary has 20 twenty shilling coins.
Substitute the value of $N_{20}$ back into Equation 1 to find $N_{50}$ .
$N_{50} = 3 \times 20$ $N_{50} = 60$
So, Mary has 60 fifty shilling notes.

The number of fifty shilling notes is $\boxed{60}$ and the number of twenty shilling coins is $\boxed{20}$ .

Given the curve y = x2 − 2x + 6, find the coordinate of the point on the curve at which the gradient is 4. Also, Mary has some money in two denominations only. Fifty shilling notes and twenty shilling

Step-by-step solution

Find the derivative of the curve to get the gradient function.

Set the gradient equal to 4 and solve for $x$ .

Substitute the value of $x$ back into the original equation to find the corresponding $y$ -coordinate.

Define variables.

Formulate equations based on the given information.

Substitute Equation 1 into Equation 2.

Solve for $N_{20}$ .

Substitute the value of $N_{20}$ back into Equation 1 to find $N_{50}$ .

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More Mathematics Questions

Key points of y = x² − 2x + 6
Point	x	y
y-intercept	0	6
Minimum	1	5