Question

1. the area of \\( \triangle a b c \\) below is 36 square centimeters. if height \\( \overline{b d} \\) is 9 centimeters, how long is the base of \\( \overline{a c} \\), in centimeters?

1. a farmer wants to double the area of his rectangular 6 - meter - by - 10 - meter garden. the 10 - meter length will be increased by 2 meters. by how many meters must the width increase?

1. on the standard \\( (x, y) \\) plane below, 1 side of the rectangle is on the axis, and the vertices of the opposite side are on the graph of the parabola given by \\( y = 7 - x^{2} \\). let \\( z \\) represent any value of \\( x \\), such that \\( 0 \leq z \leq \sqrt{7} \\). which of the following expressions, in terms of \\( z \\), represents the area of any such rectangle?

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\).

Step2: Substitute the given values into the formula

We know that \(A = 36\) square centimeters and \(height=BD = 9\) centimeters. Let the base \(AC=b\). Then \(36=\frac{1}{2}\times b\times9\).

Step3: Solve for \(b\)

First, multiply both sides of the equation \(36=\frac{1}{2}\times b\times9\) by \(2\) to get \(72 = 9b\). Then divide both sides by \(9\), so \(b=\frac{72}{9}=8\) centimeters.

Step1: Calculate the original area of the garden

The original area of a rectangle is \(A = length\times width\). Given \(length = 10\) meters and \(width = 6\) meters, so \(A_{original}=10\times6 = 60\) square meters.

Step2: Calculate the new area

The farmer wants to double the area, so \(A_{new}=2\times60=120\) square meters.

Step3: Find the new length

The new length is \(l_{new}=10 + 2=12\) meters.

Step4: Solve for the new width

Using the formula \(A = length\times width\), we have \(120=12\times w_{new}\). Solving for \(w_{new}\), we get \(w_{new}=\frac{120}{12}=10\) meters.

Step5: Calculate the increase in width

The original width was \(6\) meters. The increase is \(10 - 6=4\) meters.

Step1: Determine the length and width of the rectangle

The width of the rectangle is \(2z\) (since the rectangle is symmetric about the \(y\) - axis). The height of the rectangle is \(y = 7 - z^{2}\).

Step2: Use the formula for the area of a rectangle

The area formula for a rectangle is \(A=\text{length}\times\text{width}\). Here, \(A=(2z)\times(7 - z^{2})\).

Step3: Expand the expression

Using the distributive property \(a(b + c)=ab+ac\), we have \(A = 2z\times7-2z\times z^{2}=14z-2z^{3}=-2z^{3}+14z\).

Answer:

J. \(8\)